Methods for Estimating Location Using Signal with Varying Signal Speed

ABSTRACT

Robust methods are developed to provide bounds and probability distributions for the locations of objects as well as for associated variables that affect the accuracy of the location such as the positions of stations, the measurements, and errors in the speed of signal propagation. Realistic prior probability distributions of pertinent variables are permitted for the locations of stations, the speed of signal propagation, and errors in measurements. Bounds and probability distributions can be obtained without making any assumption of linearity. The sequential methods used for location are applicable in other applications in which a function of the probability distribution is desired for variables that are related to measurements.

RELATED APPLICATIONS

This application is a continuation of application Ser. No. 11/803,235,filed May 14, 2007, now U.S. Pat. No. 7,363,191, which is a division ofapplication Ser. No. 10/419,712, filed Apr. 20, 2003, now U.S. Pat. No.7,219,032, which claims the benefit of provisional applications Ser. No.60/373,934, filed Apr. 20, 2002, Ser. No. 60/428,994, filed Nov. 25,2002, and Ser. No. 60/451,523, filed Mar. 3, 2003.

COMPUTER PROGRAM LISTING APPENDIX

The present application incorporates by reference the material on acomputer program listing appendix. This material resides on identicalcompact disks labeled copy 1 and copy 2. All files were created on 20Apr. 2003. The names and sizes in bytes respectively of the files oneach disk are:

1. PU_(—)1m.m 3928

2. Pc_im.m 4547

3. Pcc_im.m 7224

4. Precpcs_im.m 6139

5. Ptxyz.m 6216

6. Pxyz.m 6176

7. bin_numb_to_coord.m 1371

8. boundP.m 34978

9. boundP_(—)2d.m 23932

10. boundP_(—)3d.m 34942

11. boundsposit_(—)4rec_d.m 44643

12. chkint_type.m 1014

13. comb1.m 592

14. comp1.m 769

15. confidlim.m 2327

16. drawisosigma.m 7528

17. funl.m 96

18. genarr.m 5336

19. getisosigma.m 8934

20. getrv.m 5876

21. getrvar.m 3846

22. getsrcbnds.m 6313

23. getx_rv.m 8181

24. isodiachron.m 5060

25. plot_c.m 2088

26. plot_tbounds.m 2606

27. plt_U.m 1377

28. plt_c.m 1710

29. plt_reclim.m 2258

30. plt_sbnds.m 3167

31. plt_sourcelim.m 3524

32. plt_tbnds.m 3099

33. ranint.m 792

34. removepair.m 985

35. sbnds_ttdgroup.m 3020

36. sign_chg.m 3409

37. sol_quartic_isosig_loc.m 19350

38. sol_quartic_isosig_location.m 16343

39. solve_quartic_location.m 13679

40. unpack_binmatch.m 760

41. usedirninfo.m 2625

BACKGROUND

1. Field of the Invention

In general, the present invention relates to methods for locatingobjects from measurements of distances and travel times of signals. Italso relates to methods in sequential estimation.

2. Description of Prior Art

Hyperbolic locations are typically derived by intersecting hyperboloidsfrom estimates of the differences in distances between a primary pointand pairs of stations located in space. The technique has numerousapplications including the Global Positioning System, systems to locatecell phones, systems to locate calling animals, sonar systems used inthe Navy, systems to locate lightening, underwater navigation systems,algorithms to locate earthquakes from seismic recordings, systems tolocate aircraft using electromagnetic signals, systems to locateaircrafts, missiles, and ground equipment from their acoustic emissions,systems for locating electromagnetic emissions in general includingradar and communication systems, and Positron Emission Tomography (PET)for the medical field.

Probability distributions for location have been estimated by eitherassuming that the variables, such as signal speed and station location,are linearly related to the location of the primary point (Chestnut,IEEE Transactions on Aerospace and Electronic Systems, Vol AES-18, No.2, 214-218, 1982; Dailey and Bell, IEEE Transactions on Aerospace andElectronic Systems, Vol. 32 No. 3, p. 1148-1154, 1996; Wahlberg, Mohl,and Madsen, J. Acoustical Society America, Vol. 109, 397-406, 2001) orthat the differences in distances are Gaussian random variables (Abel,IEEE Transactions on Aerospace and Electronic Systems, Vol. 26, No. 2,423-426, 1990). Both assumptions are often invalid even though they arewidely used. For example, the difference in distance is mathematicallyequal to the difference in distance between the primary point and eachstation. If the initial estimates of the station's locations areGaussian random variables, then the difference in distance is also arandom variable, but it is not Gaussian as approximated in prior art(Abel, 1990). It would be useful to be able to estimate probabilitydistributions for primary points using realistic probabilitydistributions for station coordinates, such as truncated Gaussian oruniform distributions.

Furthermore, hyperbolic locations are restricted to the case where thespeed of signal propagation is constant throughout space, in which casethe difference in distance is estimated from the difference inpropagation time of the signal. But there are many situations in whichthe speed of the signal is not constant between a primary point anddifferent stations. Examples of this occur in Global Positioning Systems(GPS), acoustic propagation in the sea, acoustic propagation in theEarth, mixed propagation cases where a signal propagates throughdifferent medium such as water and Earth, etc. U.S. Pat. No. 6,028,823by Vincent and Hu titled “GEODETIC POSITION ESTIMATION FOR UNDERWATERACOUSTIC SENSORS,” generalize hyperbolic location to one in which thespeed of sound is restricted to a function of depth only in the sea.Non-linear equations are solved for location with an iterativelinearization based on a Newton-Raphson method. One of the inventorssays that it is not possible to solve the equations analytically, so aniterative procedure is used (Vincent, Ph.D thesis, Models, Algorithms,and Measurements for Underwater Acoustic Positioning, U. Rhode Island,p. 25, 2001). He says that this iterative least squares technique“suffers from several well known disadvantages such as (1) an initialguess of object position is required to linearize the systems ofequations by expansion as a truncated Taylor series, (2) the procedureis iterative, (3) convergence is not guaranteed, and (4) the procedurehas a relatively high computational burden” (Vincent, p. 25 2001). Theprocedure in U.S. Pat. No. 6,028,823 does not provide a method forsolving for location when the speed of propagation varies in more thanone Cartesian coordinate or when the variation in speed in notpre-computed as a function of one Cartesian coordinate (U.S. Pat. No.6,388,948, Vincent and Hu). But there are many situations when the speedof the signal is not known ahead of time and also when the speed ofpropagation varies in more than one spatial coordinate both in the seaand in other environments. The method in U.S. Pat. No. 6,028,823 doesnot provide a means to solve these types of problems which commonlyoccur in GPS systems and even for systems for locating objects via theirsounds in the sea, air, or Earth.

There appears to be another technical difficulty in estimating locationerrors even when one accounts for the fact that the linear approximationis invalid. When one has more than the minimum number of stationsrequired to locate an object, the method for assessing errors does notappear to have been dealt with in the literature in a satisfying method.Schmidt's paper (Schmidt, IEEE Transactions on Aerospace and ElectronicSystems, Vol. AES-8, No. 6, 821-835, 1972), uses Monte Carlo simulationsto estimate maximum location errors of a primary point. Suppose for themoment that a mathematically unambiguous location can be achieved withthree stations for a two-dimensional geometry. Schmidt jiggles idealtravel time differences and station coordinates within their expectederrors to see how much a location changes with respect to the correctlocation. Errors are obtained by taking the largest misfit between thejiggled and correct estimate of location. When there are more than threestations, say

, he suggests using this procedure for each combination of threestations giving,

${\begin{pmatrix} \\3\end{pmatrix} = \frac{!}{{\left( { - 3} \right)!}{3!}}},$

total combinations. Each combination is referred to as a “constellation”here. Schmidt suggests using a least squares procedure to find a finalestimate of error from the results of the largest misfit from eachconstellation. He does not prove this least-squares procedure isoptimal, and indeeds states that he is not sure of any advantages inusing a least-squares procedure in this situation.

There does not appear to be a practical method for estimatingprobability distributions for hyperbolic location for the overdeterminedproblem mentioned by Schmidt (1972) when the distributions for stationlocations and estimates of differences of distance are not Gaussianrandom variables.

Bayes theorem provides a method to compute the probability distributionfor location but it requires an enormous number of computations toimplement. Suppose one wants the joint probability density function,ƒ({right arrow over (s)}, {right arrow over (r)}, {right arrow over(c)}|τ_(ij)), of the source location, {right arrow over (s)}, receiverlocations, {right arrow over (r)}, and speeds of propagation between thesource and each receiver, {right arrow over (c)}, given a measurement ofthe difference in propagation time between the source and receivers iand j, τ_(ij). Bayes theorem supplies the desired result,

$\begin{matrix}{{{f\left( {\overset{\rightarrow}{s},\overset{\rightarrow}{r},{\overset{\rightarrow}{c}\tau_{ij}}} \right)} = \frac{{f\left( {{\tau_{ij}\overset{\rightarrow}{s}},\overset{\rightarrow}{r},\overset{\rightarrow}{c}} \right)}{\pi \left( {\overset{\rightarrow}{s},\overset{\rightarrow}{r},\overset{\rightarrow}{c}} \right)}}{\int{{f\left( {{\tau_{ij}\overset{\rightarrow}{s}},\overset{\rightarrow}{r},\overset{\rightarrow}{c}} \right)}{\pi \left( {\overset{\rightarrow}{s},\overset{\rightarrow}{r},\overset{\rightarrow}{c}} \right)}{\overset{\rightarrow}{s}}{\overset{\rightarrow}{r}}{\overset{\rightarrow}{c}}}}},} & (1)\end{matrix}$

in terms of the conditional probability, ƒ(τ_(ij)|{right arrow over(s)}, {right arrow over (r)}, {right arrow over (c)}), of obtaining thedata, τ_(ij), given the locations of the source and receivers, andspeeds of propagation, and in terms of the prior joint distribution,π({right arrow over (s)}, {right arrow over (r)}, {right arrow over(c)}). If the distributions on the right side could be evaluatedanalytically, then evaluating Eq. (1) would possibly provide a practicalsolution to the problem. One could introduce new data, and keep findingbetter estimates of the distribution on the left given updateddistributions on the right. It appears difficult to find analyticalsolutions for the distributions on the right because the variables arerelated in nonlinear ways from which progress appears to be at astandstill. Brute force evaluation of the distributions on the rightalso appears computationally impractical because of the highdimensionality of the joint distributions. In hyperbolic location, forexample, for R receivers, there are three variables for the Cartesiancoordinates of the source and each receiver, as well as a speed ofpropagation making a total of 3+3R+1=3R+4 variables. For R=5, one wouldneed to estimate the joint probability density functions of 19variables, for each introduced datum (a 19 dimensional space). Supposeeach variable is divided into ten bins. Accurate estimation of the jointdistribution requires a reliable probability of occurrence in 10¹⁹ binsin the example given above. Since this is not practical to compute,Bayes theorem does not offer a practical means to compute probabilitydistributions of location for many cases of interest.

It is useful to be able to estimate probability distributions forvariables of interest that occur in non-linear situations or when apriori distributions are not Gaussian. The location of a primary pointfrom measurements of the difference in arrival time at two or morestations is an example of a nonlinear problem when variables may not bedescribed as being Gaussian. When variables are Gaussian, a Kalmanfilter can be used to estimate their probability distributions (Gelb,Applied Optimal Estimation, p. 105-107, M.I.T. Press, Cambridge, Mass.1996). When the Kalman filter is not applicable to a problem, one canattempt to find solutions by linearizing equations and iterating tooptimize an objective function. This approach may not converge to thecorrect solution if the initial guess is not close to the correctanswer. There is a need for a procedure to estimate probabilitydistributions for non-linear systems or for systems in which thevariables are not Gaussian.

For ellipsoidal locations, there does not appear to be a method forobtaining probability distributions for the primary point when using thesums of distances or the sums of travel times between two points and aprimary point without making a linear approximation or without assumingsome or all of the variables are Gaussian random variables. Furthermore,when one uses measurements of travel time to estimate the sums ofdistances, ellipsoidal location methods assume that the speed ofpropagation is constant. Ellipsoidal locations are strictly invalid whenthe speed of propagation varies between the stations. It would be usefulto have a method to obtain locations when the speeds of propagation arenot the same between each instrument.

There do not appear to be methods to estimate probability distributionsfor initially unknown locations of a primary point and for locations ofthe stations and the speed of propagation without recourse tolinearizing approximations and assumptions of Gaussian distributions.For example, Dosso and Collison (Journal of the Acoustical Society ofAmerica, Vol. 111, 2166-2177, 2002) linearize equations betweenunderwater station locations and the location of a acoustic source in aproblem dealing with hyperbolic location. They assume the location ofthe source is approximately known and that prior probabilitydistributions for the data and locations of source and stations areGaussian. But it would be useful to have a robust method to estimatethese distributions without recourse to Gaussian statistics and withoutmaking a linear approximation, which may be inaccurate and which maylead to a solution in a non-global minimum of a cost function.

SUMMARY

In accordance with the present invention, methods are developed forestimating probability distributions for location and for the variablesthat affect location such as the coordinates of stations and the speedsof signal propagation. The methods are robust, practical to implement,and admit nonlinear relationships between data and variables of interestwithout recourse to any linear approximation. The estimation methods ofthe present invention have applications in many other fields.

OBJECTS AND ADVANTAGES

The invention provides a method for estimating probability distributionsfor variables that are functionally related to data. The priordistribution of the variables can be realistic and need not be Gaussian.The function relating the variables to the data can be linear ornonlinear. When it is nonlinear, it is not necessary to use a linearapproximation to implement the method. For location problems, theinvention yields probability distributions for the location of theprimary point as well as for the coordinates of stations and the speedsof signal propagation when the data consist of travel times ordifferences in travel times. Thus, the method provides a means toconduct self surveying and tomography, with accompanying probabilitydistributions, at the same time as providing distributions for location.

The invention provides a method for locating a primary point fromdifferences in travel time when the speed of propagation can bedifferent for any path. To facilitate an understanding of the geometryof this situation, which is no longer hyperbolic, a new geometricalobject called an “isodiachron” is introduced. The word is derived fromthe Greek words “iso”, for same, “dia”, for difference, and “chron”, fortime. The surface is one along which the locus of points has the samedifference in travel time between two stations. This reduces to ahyperboloid only if the signal speed is spatially homogeneous. Despitethe fact that the literature states that there is no analytical solutionfor location in this situation (Vincent, 2001), an analytical solutionfor location is derived as part of the present invention. This solutionis not constrained to permit only variations of speed with depth, butallows variations to occur for any and all coordinates. The invention ofthe analytical solution is efficient enough to allow estimation ofprobability distributions for any variables in the problem.

The inventions for estimating location reveal, perhaps for the firsttime, how badly the ubiquitous linear approximation is for estimatingerrors for hyperbolic locations in many cases of interest. In fact, theinvention demonstrates that probability distributions for location aresometimes overestimated by ten to one-hundred times compared with thefully nonlinear estimates obtained from this invention.

As said previously, the invention provides a different method forobtaining probability distributions of location than obtained elsewhere(Chestnut, 1982: Dailey and Bell, 1996; Wahlberg, Mold, and Madsen,2001). In particular, when one has more than the minimum number ofnecessary stations to obtain the location of a primary point, (theoverdetermined problem), the new method called “Sequential BoundEstimation” is used to obtain probability distributions.

The invention provides means for estimating probability distributions ofellipsoidal or “isosigmachronic” (defined below) location without makinguse of a linear approximation or an approximation that the data andother variables are Gaussian. Instead, the invention providesprobability distributions of location based on a priori distributions ofthe speeds of the signal, locations of stations, and errors in themeasurements of the travel times of the signal.

In order to facilitate an analytical solution for location when thespeed of signal propagation differs between the primary point and someof the stations, a new geometrical object is introduced called an“isosigmachron.” It is derived from the Greek words “iso”, for same,“sigma” , for sum, and “chron”, for time. The surface is one along whichthe locus of points has the same sum in travel times between twostations and a primary point. The isosigmachron reduces to an ellipsoidonly if the signal speed is spatially homogeneous.

The invention provides means for estimating probability distributionsfor problems in spherical navigation through use of Sequential BoundEstimation.

These as well as other novel advantages, details, embodiments, featuresand objects of the present invention will be apparent to those skilledin the art from the following detailed description of the invention, theattached claims and accompanying drawings, listed hereinbelow, which areuseful in explaining the invention

DRAWING FIGURES

In the text which follows and in the drawings, wherein similar referencenumerals denote similar elements throughout the several views thereof,the present invention is explained with reference to illustrateembodiments, in which:

FIG. 1 Flowchart for Sequential Bound Estimation.

FIG. 2. Flowchart for obtaining a function of the probabilitydistribution for the hyperbolic location of a primary point and afunction of the probability distribution for other variables that affectthe location of a primary point.

FIG. 3. A,B: Two dimensional hyperbola (dashed) compared withtwo-dimensional isodiachron (solid). Locations of the two receivers(asterisks) are at Cartesian coordinates (−1,0) and (1,0). The averagespeeds of propagation between the source and receivers one and two are330 and 340 m s⁻¹ respectively. The lag, τ₁₂ is 0.0015 s. The hyperbolais computed for a speed of 330 m s⁻¹. C,D: Same except the lag is−0.0015 s.

FIG. 4. Flowchart for obtaining analytical solution for location ofprimary point using isodiachronic or isosigmachronic location.

FIG. 5. Flowchart for iterating analytical solution of isodiachronic andisosigmachron location when the affects of advection significantlyaffect the travel time of a signal between a primary point and astation.

FIG. 6. Flowchart for estimating probability distribution for anyvariable related to isodiachronic location.

FIG. 7 The receiver and source locations for arrays one and two.

FIG. 8 100% confidence limits for source location as a function of thenumber of receiver constellations used from array one (FIG. 7). Thereare five ways of choosing four receivers from five total withoutreplacement. A receiver constellation consists of one of the choices offour receives. As more constellations are used to locate the source, thebounds for the source's location decrease monotonically. The lines joinresults from different numbers of constellations.

FIG. 9 Probability functions for source location from array one (FIG. 7,top), calculated from the non-linear method in this specification.Receiver constellation 2 is receivers 1, 2, 3, and 5 (FIG. 7, top).Receiver constellation 4 is receivers 1, 3, 4, and 5.

FIG. 10 Same as FIG. 8 except for array two in FIG. 7 and there are 35ways of choosing 4 receivers from a total of 7 without replacement.

FIG. 11 Probability functions for source location from array two (FIG.7, bottom) using the non-linear method in this specification. Receiverconstellations 7, 4, and 26 are composed from receivers {1, 2, 4, 7},{1, 2, 3, 7}, and {2, 3, 6, 7} respectively.

FIG. 12 Probability functions for the Cartesian (x, y, z) coordinates ofreceiver two (Table III) as determined from Sequential Bound Estimation.The y and z coordinates are set to zero by definition. The correctanswer for the x coordinate is 19.76±0.05 m.

FIG. 13. Horizontal locations of 6 receivers and 10 sources (circles)used to simulate estimation of the spatially homogeneous component ofthe field of sound speed. The receivers have elevations of 0, 0, 0,−0.02, −0.43, and −5 m respectively.

FIG. 14. Geometry for an isosigmachron. The travel time from s toprimary point P plus from P to station r_(i) is t₀+t₁, a known quantity.The effective speed of propagation between s and P, c₀, differs from theeffective speed from P to r_(i), c_(i). The locus of points along whichthe sum of travel times, t₀+t₁ is constant defines an isosigmachron. Theprimary point P is one of the points on the isosigmachron.

FIG. 15. Comparison of two-dimensional isosigmachron with ellipse. Theisosigmachron is drawn for stations at Cartesian coordinates (±1, 0)(asterisks), and for speeds c₀=3 m/s and c₁=2 m/s respectively, assumingthe travel time is T=1.7 s from the station at x=−1 to the isosigmachronand back to x=+1. The ellipse is drawn for the same stations and traveltime but assuming the speed of propagation is the average of c₀ and c₁.

FIG. 16. Cloud of locations of the target after using travel time fromsource to primary point plus primary point to receiver 1 (R1) and usingthe information at R1 for the direction to the primary point.

FIG. 17. Same as FIG. 16 except Sequential Bound Estimation is used tocombine the travel time and directional data from receiver 1 andreceiver 2 (R2).

FIG. 18. Flowchart for estimating probability distribution for anyvariable related to isosigmachronic location.

FIG. 19. Flowchart for estimating bounds and probability distributionsof location of a primary point for ellipsoidal or isosigmachronicgeometries when both the direction and time of the incoming signal areestimated at each station from the signal emitted at a source.

TABLE I Cartesian coordinates of arrays one and two (FIG. 7) and theirstandard deviations. Receiver one's location is defined to be the originof the Coordinate system, and so has zero error. The y coordinate ofreceiver two is defined to be at y=0, and thus has zero error.

TABLE II 68% confidence limits for source locations for the two arrayslisted in TABLE I. Results are given for linear and nonlinear analysis.

TABLE III Cartesian (x,y,z) coordinates for the five receivers shown inFIG. 3 of Spiesberger, Journal Acoustical Society America, Vol. 106,837-846, 1999. Non-zero locations are measured optically within ±0.05 m.

TABLE IV 68% confidence limits for Standard Error Model L1 C/A no SA forthe Global Positioning System (Table 2 inhttp://www.edu-observatory.org/gps/check_accuracy.html). Errors in somecases are translated into equivalent errors in the distance, L, ofpropagation using L=ct where the speed of light is c and t is time.

TABLE V Eight GPS receivers used for isodiachronic location. These areactual satellite locations near Philadelphia for a certain moment intime.

TABLE VI 68% confidence limits for the precise error model,dual-frequency, P(Y) code for no SA for the Global Positioning System(Table 4 in http://www.edu-observatory.org/gps/check_accuracy.html).Errors in some cases are translated into equivalent errors in thedistance, L, of propagation using L=ct where the speed of light is c andt is time.

TABLE VII 68% confidence limits for the precise error model,dual-frequency, P(Y) code for no SA in differential model for the GlobalPositioning System (Table 4 inhttp://www.edu-observatory.org/gps/check_accuracy.html). Errors in somecases are translated into equivalent errors in the distance, L, ofpropagation using L=ct where the speed of light is c and t is time.

TABLE VIII Cartesian (x, y) locations (m) of the source, primary point,and five receivers for the ellipsoidal and isosigmachronic locationexperiments.

TABLE IX Times (s) for the inventor to walk from the source to primarypoint, and then from the primary point to each of five receivers (TABLEVIII) as measured by a stopwatch.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

For a better understanding of the location techniques, reference may behad to the following detailed description taken in conjunction with theappended claims and accompanying drawings.

I. Sequential Bound Estimation

A concise description of Sequential Bound Estimation is made first withsome examples provided. Further understanding of the workings of thealgorithm are provided later through many simulated and experimentalexamples.

Referring to FIG. 1, Sequential Bound Estimation starts by determiningprior bounds for arguments of a primary function and bounds for theprimary function itself 10. The term “bounds” in this context means theminimum and maximum value of an argument of the function or the minimumand maximum value of the primary function. If an argument of the primaryfunction is a quantity other than a scaler, then bounds are establishedfor each of the components. For example, the bound for thethree-dimensional vector wind field consists of six numbers, an upperand lower bound for each of the (x, y, z) Cartesian components of thewind. The term “bound” refers to either the upper or lower value for ascaler quantity. In many location experiments, one establishes priorbounds for the location of the primary point, the locations of thestations, the speeds of propagation, and errors in the measurements.

As shown in the computer program called “sbnds_ttdgroup.m” in theAppendix, one coordinate of the location of a source can depend onanother variable, such as a different location coordinate; in otherwords, bounds need not be constants. If location is expressed in twodimensions (such as x-y coordinate pairs), for example, the bounds forthe output location coordinate can depend on the value of the othercoordinate. The computer program “sbnds_ttdgroup.m” divides x-y spaceinto rectangles forming a grid and describes the bounds on the xlocation variable by identifying possible (i.e., in-bound) gridrectangles, allowing x to have different upper and lower boundsdepending on the value of y.

In the context of this invention, the term “variables” is used to referto the set of arguments of the primary function as well as values of theprimary function itself. For isodiachronic location, the variables arethe noisy measurements, the speeds of propagation between each stationand the primary point, the winds, the noisy estimates for the locationsof the stations, and the location of the primary point. Next, priorprobability distributions are assigned to all arguments of the primaryfunction. The distributions confine the arguments within theirrespective bounds 20. The primary function is excluded 20 because it isdetermined by its arguments.

A probability distribution is some complete way of describing thestatistics of a variable. A probability distribution can be aprobability density function. A cumulative density function, the Fouriertransform of a probability density function, or other equivalentrepresentations.

Using the first data, realizations of values of the primary function aregenerated from realizations of its arguments with a Monte-Carlotechnique 30. Only values of the primary function within its priorbounds are retained. For example, some of the realizations may give riseto values of the primary function beyond its prior bounds. These values,and the realizations of arguments that generated these values areexcluded from further consideration. For example, there may be sets ofrandom locations for the receivers, speeds of propagation, winds, andmeasurement errors that lead to locations of a primary point that arebeyond where the primary point could lie. Such exclusion can occur ifone is locating an airplane and a particular set of arguments places itslocation underground.

The next step involves finding posterior bounds for the variables in theproblem from the set that remain after using the Monte-Carlo technique40. For a location problem, one can find the spatial bounds for aprimary point, and bounds for each of the other variables that are usedto generate the location such as the speed of propagation, the winds,the errors in the measurements, and the locations of stations.

The next set of data are used to repeat the process through one of tworoutes. If one desires to use a model to transition the posterior boundsfor some or all of the variables, it can be done 60. The posteriorbounds that are transitioned become the prior bounds for the next data20. For a location problem, one may want to use a posterior bound forthe x coordinate of a stationary receiver to be the prior bound for thex coordinate for use with the next data when this transition leads to asmaller interval for x. Or to take another example, one may want to usea model to change the posterior bounds to be consistent with the nextset of data. In the location problem, suppose the first data are takenat 1 PM in the afternoon, and the posterior bounds for the speed ofsound are between 330 and 333 m/s. If the next set of data are taken at2 PM, one can use a model to expand the posterior bounds to 325 to 336m/s corresponding to a growing uncertainty of the speed of sound withtime. These become the prior bounds for the data taken at 2 PM.

Alternatively, one may desire to not transition posterior bounds, andproceed with the prior bounds for use with the next data 50. This can bedone if the prior bounds are acceptable for use with the next data.

An important part of Sequential Bound Estimation is the assignment ofthe prior probability distribution after using the first data 20.Instead of transitioning the shapes of the distributions themselves fromthe first to second data, one approximates the prior distributions forthe second and later data with a prior distribution that yields valuesfor the variables only between their prior bounds. Bayes theorem andKalman filters do not do this at all. Instead, Bayes theorem and Kalmanfilters have statistically rigorous procedures for transitioning theprobability distributions. But as described before, there may be toomany computations involved with Bayes theorem for practical use, andKalman Filters usually deal only with simple distributions of variableslike Gaussian distributions, that may not be at all realistic.Sequential Bound Estimation does not evolve the shape of the probabilitydistribution for use with the next data. Instead Sequential BoundEstimation evolves bounds for the variables. A cautious practitioner maywant to impose a distribution of maximum ignorance such as one that isuniform between an argument's associated bounds. As will be seen laterin simulations and experiments, this method of assigning priorprobability distributions is useful and is one of the features thatmakes Sequential Bound Estimation a robust algorithm.

The algorithm proceeds sequentially until there are no more data. Atthis point, final bounds are estimated for the variables 70.

The procedure for establishing final bounds 70 depends on the variableand whether bounds for that variable were transitioned. There are fourpossibilities. Two possibilities occur because a posterior bound may ormay not be transitioned. Two other possibilities occur because thestatistics of the quantity that is modeled by a variable, may be afunction of the sequence step.

First, suppose posterior bounds for a variable are not transitioned andsuppose the quantities' statistical properties do not change. Forexample, in a navigation problem, the quantity could be the x coordinateof a receiver that does not move. The variable would represent the xcoordinate of a receiver. Let the upper and lower posterior bound forthe variable for sequential step j be denoted as,

{right arrow over ({circumflex over (V)}_(j)   (2)

{right arrow over ({hacek over (V)}_(j)   (3)

respectively. A final bound 70 is given by,

{right arrow over ({circumflex over (V)}=min{{circumflex over (V)}_(k)},kε{1, 2, 3, . . . , J}  (4)

{right arrow over ({hacek over (V)}=max{{hacek over (V)}_(p)}, pε{1, 2,3, . . . , J},   (5)

where there are at least two entries in each of the sets containing{circumflex over (V)}_(k) and {hacek over (V)}_(p) respectively. Thebest estimate of the final bounds are obtained in Eqs. (4,5) by usingall J sets of data.

Second, suppose the posterior bounds for a variable are transitioned andsuppose the quantities' statistical properties do not change. Then theposterior bound becomes the prior bound at the next step. For example,the x coordinate of a receiver may not change. Then the final bounds 70for x are obtained from the J'th sequential step, namely {right arrowover ({circumflex over (V)}_(J) and {right arrow over ({hacek over(V)}_(J).

Third, suppose the posterior bounds for a variable are not transitionedand suppose the quantities' statistical properties do change. This maynot be the wisest way to implement this algorithm, but it is possible torun the algorithm in this manner. This means that the bounds for thevariable are the prior bounds established at step 10. Taking the sameexample as before, the x coordinate of a receiver changes with time. Inthis case, the prior bounds for this coordinate must be large enough toencompass all possible values of x. If it is somehow known that thechange in x is very small for all sequential steps compared with itsprior bounds, one can choose a final bound from Eqs. (4,5). Otherwise,one can choose the final bounds for x from the posterior bounds 40 atthe sequential step desired.

Fourth, suppose the posterior bounds for a variable are transitioned andsuppose the quantities' statistical properties do change. For example,suppose the x coordinate of a receiver changes with time. The variablewould represent the x coordinate of a moving receiver. In this case, onemust decide at which sequential step one desires a final bound. If thebounds for each step are small compared with the change in x, onespecifies sequential step j and the final bounds can be obtained from{right arrow over ({circumflex over (V)}_(j) and {right arrow over({hacek over (V)}_(j). On the other hand, it may be known ahead of timethat the change in x may be very small compared with its posteriorbounds from the sequential steps, so one may get excellent estimatesfrom Eqs. (4,5). So in this case, it may be seen that judgment can berequired for obtaining final bounds 70.

When it is desired to obtain further probability distributions of avariable, the procedure in 80 is implemented.

One function of a probability distribution are the variables' bounds.Other commonly used functions are 95% confidence limits, maximumliklihood values, median values, average values, or the identityfunction yielding the probability distribution itself, etc. A “functionof a probability distribution” simply refers to a function applied to aprobability distribution to obtain a desired summary of the statisticalproperty of that variable.

To implement 80 it is first necessary to compute final bounds for thevariable of interest for the sequential step desired 70. Empiricalfunctions of the distributions are derived from the Monte-Carlorealizations of each variable that were previously computed. To showwhat is done, we examine the same four cases as above.

First, suppose the posterior bounds for a variable are not transitionedand suppose the quantities' statistical properties do not change. Forexample, in a navigation problem, the quantity could be the x coordinateof a receiver that does not move. The final bounds, given by Eqs. (4,5),are used to go back and retain only realizations of x between {rightarrow over ({circumflex over (V)} and {circumflex over ({hacek over (V)}for each desired sequential step. One then chooses a criterion for thebest function of the distributions. For example, realizations fromsequential step p, truncated between {right arrow over ({circumflex over(V)} and {circumflex over ({hacek over (V)}, may have the smallest 95%confidence intervals, and those are the outputs 80.

Second, suppose the posterior bounds for a variable are transitioned andsuppose the quantities' statistical properties do not change. Then thefinal bounds 70 for x are obtained from the J'th sequential step, namely{right arrow over ({circumflex over (V)}_(J) and {right arrow over({hacek over (V)}_(J). The realizations from step J automatically fallwithin these posterior bounds. One then chooses a function of thedistribution from those realizations of x from step J to produce thedesired function of the distribution.

Third, suppose the posterior bounds for a variable are not transitionedand suppose the quantities' statistical properties do change. Taking thesame example as before, suppose the x coordinate of a receiver changeswith time. If it is somehow known that the change in x is very smallcompared with its prior bounds for all sequential steps, one can choosea final bound from Eqs. (4,5). These final bounds, are used to go backand retain only realizations of x between {right arrow over ({acute over(V)} and {right arrow over ({hacek over (V)} for each desired sequentialstep. One then chooses the step p yielding the best values for thefunction of the probability distribution of x 80. On the other hand, ifx changes significantly compared with the interval of its prior bounds,one can choose the posterior bounds for x at step 40 corresponding tothe sequential step desired, and obtain the desired function of itsprobability distribution from the realizations at that step.

Fourth, suppose the posterior bounds for a variable are transitioned andsuppose the quantities' statistical properties do change. For example,the x coordinate of a receiver changes with time. If the posteriorbounds for each step are small compared with the change in x, theposterior bounds in 40 can be the final bounds and the desired functionof x's distribution 80 is derived from the realizations of x 30. If thechange in x is very small compared with the its posterior bounds 40, thebounds, {right arrow over ({circumflex over (V)} and {circumflex over({hacek over (V)}, can be the final bounds 70. One then uses therealizations of x at 40 within {right arrow over ({circumflex over (V)}and {circumflex over ({hacek over (V)} at each sequential step, and thefunction of the probability distribution of x is computed for each ofthe sequential steps. This function, for each step j, yielding the bestvalue, is output 80

It is important to note that Sequential Bound Estimation does notattempt to combine different distributions for a particular variable toobtain an optimal distribution for that variable. Bayes rule provides ameans for combining different data to obtain a single distribution. Butbecause there are so many computations required to implement Bayestheorem for many problems of interest, Sequential Bound Estimation canbe used instead because it provides an efficient and usefulapproximation for obtaining probability distributions.

Like other sequential algorithms, Sequential Bound Estimation can be runbackward to improve the quality of the estimates. For example, supposeafter using data set 2 from the forward direction, that the posteriorbounds for variables before transitioning 60 are denoted {circumflexover (V)}₂ ⁺ and {hacek over (V)}₂ ⁺. For the reverse run, let theposterior bounds immediately after using data set 2 be denoted{circumflex over (V)}₂ ⁻ and {hacek over (V)}₂ ⁻. The best upper andlower bounds from the forward and backward runs are given by the minimumof {{circumflex over (V)}₂ ⁺, {hacek over (V)}₂ ⁻} and the maximum of{{circumflex over (V)}₂ ⁺, {hacek over (V)}₂ ⁻} respectively. One simpleway to implement Sequential Bound Estimation is to place the Jsequential data in the order d₁, d₂, d₃, . . . , d_(J), d_(J−1),d_(J−2), d_(J−3), . . . , d₁ and run the algorithm for this ordering.

Sequential Bound Estimation has the desirable feature in that it candetect problems with the data, or assumptions about the variables, andso save a user from believing everything is ok when it is not. Forexample, suppose data are accidentally mis-ordered and prior andposterior bounds are found for the x coordinate of a non-movingreceiver. Further suppose that the posterior bounds are nottransitioned. Sequential Bound Estimation may then end up with aposterior upper bound for x for one datum that is less than a posteriorlower bound for x from a different datum. This is impossible andSequential Bound Estimation cannot produce a feasible solution becausesomething is wrong with its inputs. The algorithm halts at the pointwhen final bounds are estimated 70. Another possibility occurs when thealgorithm yields few or no realizations of a variable within itsassigned bounds 40. If there are no realizations within prior bounds,the algorithm needs to stop. Stopping the algorithm is desirable forquality control so one can go back and figure out what went wrong. Notall algorithms have this property. For example, linear and nonlinearleast squares produce answers regardless of many problems with data. Forthe purpose of catching problems early, it may be established thatsomething is wrong if fewer than a minimum number of realizations areobtained for the value of a primary function, where the minimum numbercan be zero or some small positive value compared with the number ofMonte-Carlo realizations.

It is important to note that Sequential Bound Estimation is a robustalgorithm. Its results are insensitive to nonlinearities in the systemand to details within the distributions of the variables, especiallywhen distributions are chosen to have the least information as foundwith uniform distributions 20.

II. Probability Distributions for Hyperbolic Location

In this related invention, hyperbolic locations are estimated fromdifferences in distance or differences in travel time to inferdifferences in distance, given a speed of signal propagation. The use ofthe word “data” in the context of hyperbolic location means eitherdifferences in distance or differences in travel time, depending on whattype of data are measured.

The word “variables” in this discussion refers to the data, estimates ofa location, estimates for the speed of propagation, and estimates forthe locations of stations.

The invention provides a method to obtain any function of theprobability distribution for the location of a primary point and for anyother variable that functionally influences the location of the primarypoint such as the location of a station. The method for accomplishingthese goals is related to Sequential Bound Estimation as will bedescribed below.

A constellation of stations is defined to be the minimum number requiredto obtain an unambiguous solution when all constraints, if any, areimposed on any of the variables in the problem. An unambiguous solutionis one in which there is only one location of a primary point if one hadperfect and noiseless estimates for the data, and the locations of thestations. For example, in three spatial dimensions without any a priorifinite bounds for the location of a primary point, a constellationconsists of four stations for primary points in some locations, and offive stations if primary points occur in other locations (Spiesberger,J. Acoustical Society America, Vol. 109, 3076-3079. 2001). With theintroduction of bounds or constraints of other types, the number ofstations in a constellation may decrease.

Referring to FIG. 2, data are collected from a constellation of stations90. Prior bounds are established for the primary point and for allvariables that affect its location 100.

An optional database of pre-computed simulated data may be used toquickly compute the spatial bounds encompassing the primary point 150.The smallest bounds (between the prior set and those from the database)are used to establish prior bounds of the probability distributions forthe primary point 110. The database is computed using realizations ofdata from a Monte-Carlo technique. The region of space within the priorbounds for the primary point is subdivided into cells givinghypothetical locations of the primary point. The primary point isassigned a uniform distribution within each cell, but anotherdistribution could be used. The Monte-Carlo technique is used togenerate realizations of data given realizations for the location of theprimary point in each cell, realizations of the winds, speed ofpropagation if using travel time differences, and locations of thestations. The minimum and maximum values of the simulated data arestored in a database for each cell. Bounds for the primary point arecomputed by finding the cells having minimum and maximum values ofsimulated data that bound each corresponding data within measurementerrors.

With or without the optional database, probability distributions areestablished for all variables except the primary point 110. Thesedistributions may be uniform if one desires maximum uncertainty. Forexample, one may establish that the x coordinate of station three isuniformly distributed between 10 and 10.8 m.

The Monte-Carlo technique is used to generate realizations of theprimary point using an analytical solution for location 120.Realizations lying outside the initial bounds for the primary point areexcluded. It is important to link each realization of variables to itsassociated realization of the primary point. At the end of this step120, one has sets of realizations of variables each linked with thelocation of a primary point. Any function of the probabilitydistribution for the primary point or other variable may be computedfrom these sets of realizations 140.

The procedure for estimating posterior bounds and functions ofprobability distributions from realizations of any variables is likethat done in Sequential Bound Estimation except one is not using asequence of data. So the method in this related invention could becalled Bound Estimation.

The speeds of propagation in this method may include the effects ofadvection, such as winds, on the time it may take a signal, such assound, to propagate between two points. Because hyperbolic locationsassume a single speed, the affects of advection can be partly accountedfor by increasing bounds on the signal speeds without advection byadditive amounts of ±|U| where |U| is the magnitude of the largestadvection in the problem.

Like Sequential Bound Estimation, this algorithm can detect problemswith the data, or assumptions about the variables by yielding nosolutions for a variable that are consistent with the assumptions. Thealgorithm must halt if an inconsistency is found 120. One can alsoinform the user and possibly halt the algorithm if there are fewer thana minimum number of acceptable realizations within prior bounds.

III. Locating Sounds Using Four or Five Receivers with InhomogeneousSound Speed Field

Instead of least-squares, a Monte-Carlo technique is used to estimate aprobability distribution of location from each constellation ofstations. For a constellation, the speed of propagation, winds, andlocations of receivers are treated as random variables with someprobability distribution. The probability distribution can be obtainedfrom theory, data, or a guess. When guessing, it may be advantageous tolet the probability distribution have the largest possible bounds withthe most ignorance so as to not overestimate the accuracy of a location.For example, suppose one believes the x location of a primary point isat 10 m with an error of two centimeters. Then the associatedprobability distribution for this parameter can be taken to be auniformly distributed random variable in the interval [9.8, 10.2] m.With the a priori probability distribution, the various values of thesound speed, winds, and station locations yield possible locations of aprimary point that occupy a cloud in three-dimensional space. The actualprimary point must lie within the intersection of the clouds fromdifferent constellations. The final probability distribution for theprimary point is estimated from the distributions within theconstellations. The density function has the benefit of being able toyield many useful values such as an average location, maximum likelihoodlocation, and any desired confidence limit, including upper and lowerbounds.

In much of what is discussed below, the primary point is treated as asource of acoustic waves and these waves are picked up at receivers. Themathematics is the same for electromagnetic propagation, except thatwinds and currents do not affect the speed of electromagneticpropagation as they do for acoustic propagation. The mathematics is alsothe same if one is trying to locate a receiving point from differencesin propagation time from pairs of synchronized transmitters, such asoccurs for some location techniques for cellular phones and GPS.Hyperbolic and isodiachronic location techniques apply equally to thesesituations.

Instead of restricting the speed of sound to be a constant, it isallowed to have a different average value between the source and eachreceiver. Although five receivers are required in general to locate asource in three dimensions in a homogeneous sound speed field, there arespatial regions where only four are required (Schmidt, 1972). Whenambiguous solutions occur with four receivers, the ambiguity is resolvedwith the travel time difference between the first and fifth receiver.The presentation below thus starts with a minimum constellation of fourreceivers.

The distance between the ith receiver at {right arrow over (r)}_(i) anda source at {right arrow over (s)} is ∥{right arrow over (r)}_(i)−{rightarrow over (s)}∥ so we have ∥{right arrow over (r)}_(i)−{right arrowover (s)}∥²=c_(i) ²t_(i) ²=c_(i) ²(τ_(i1)+t₁)². The average speed andtravel time of sound between the source and receiver i are c_(i) andt_(i) respectively. τ_(i1)≡t_(i)−t₁ and {right arrow over (s)} is acolumn vector with Cartesian coordinates (s_(x), s_(y), s_(z))^(T) whereT denotes transpose. Putting the first receiver at the origin of thecoordinate system, one subtracts the equation for i=1 from i=2, 3, and 4to get,

∥{right arrow over (r)}_(i)∥²−2{right arrow over (r)} _(i) ^(T) {rightarrow over (s)}=c _(i) ²τ_(i1) ²+2c _(i) ²τ_(i1) t ₁ +t ₁ ²(c _(i) ² −c₁ ²).

This simplifies to,

R{right arrow over (s)}=½{right arrow over (b)}−t ₁{right arrow over(ƒ)}−t ₁ ² {right arrow over (g)},   (6)

where,

$\begin{matrix}{{{R \equiv \begin{pmatrix}{r_{2}(x)} & {r_{2}(y)} & {r_{2}(z)} \\{r_{3}(x)} & {r_{3}(y)} & {r_{3}(z)} \\{r_{4}(x)} & {r_{4}(y)} & {r_{4}(z)}\end{pmatrix}};{\overset{\rightarrow}{b} \equiv \begin{pmatrix}{{{\overset{\rightarrow}{r}}_{2}}^{2} - {c_{2}^{2}\tau_{21}^{2}}} \\{{{\overset{\rightarrow}{r}}_{3}}^{2} - {c_{3}^{2}\tau_{31}^{2}}} \\{{{\overset{\rightarrow}{r}}_{4}}^{2} - {c_{4}^{2}\tau_{41}^{2}}}\end{pmatrix}};{\overset{\rightarrow}{f} \equiv \begin{pmatrix}{c_{2}^{2}\tau_{21}} \\{c_{3}^{2}\tau_{31}} \\{c_{4}^{2}\tau_{41}}\end{pmatrix}}},} & (7)\end{matrix}$

and,

$\begin{matrix}{{\overset{\rightarrow}{g} \equiv {\frac{1}{2}\begin{pmatrix}{c_{2}^{2} - c_{1}^{2}} \\{c_{3}^{2} - c_{1}^{2}} \\{c_{4}^{2} - c_{1}^{2}}\end{pmatrix}}},} & (8)\end{matrix}$

and where the Cartesian coordinate of {right arrow over (r)}_(i) is(r_(i)(x), r_(i)(y), r_(i)(z)). Eq. (6) simplifies to,

$\begin{matrix}{{\overset{\rightarrow}{s} = {{R^{- 1}\frac{\overset{\rightarrow}{b}}{2}} - {R^{- 1}\overset{\rightarrow}{f}t_{1}} - {R^{- 1}\overset{\rightarrow}{g}t_{1}^{2}}}},} & (9)\end{matrix}$

that can be squared to yield,

$\begin{matrix}{{{{\overset{\rightarrow}{s}}^{T}\overset{\rightarrow}{s}} = {{\overset{\rightarrow}{s}}^{2} = {\frac{a_{1}}{4} - {a_{2}t_{1}} + {\left( {a_{3} - a_{4}} \right)t_{1}^{2}} + {2a_{5}t_{1}^{3}} + {a_{6}t_{1}^{4}}}}},} & (10)\end{matrix}$

where

α₁≡(R ⁻¹ {right arrow over (b)})^(T)(R ⁻¹ {right arrow over (b)})

α₂≡(R ⁻¹ {right arrow over (b)})^(T)(R ⁻¹{right arrow over (ƒ)})

α₃≡(R ⁻¹{right arrow over (ƒ)})^(T)(R ⁻¹{right arrow over (ƒ)})

α₄≡(R ⁻¹ {right arrow over (b)})^(T)(R ⁻¹ {right arrow over (g)})

α₅≡(R ⁻¹{right arrow over (ƒ)})^(T)(R ⁻¹ {right arrow over (g)})

α₆≡(R ⁻¹ {right arrow over (g)})^(T)(R ⁻¹ {right arrow over (g)})

(11)and R⁻¹ is the inverse of R. A solution for t₁ is obtained bysubstituting,

∥{right arrow over (s)}∥²c₁ ²t₁ ²,   (12)

for ∥{right arrow over (s)}∥² in Eq. (10) to yield a quartic equation int₁,

$\begin{matrix}{{{{a_{6}t_{1}^{4}} + {2a_{5}t_{1}^{3}} + {\left( {a_{3} - a_{4} - c_{1}^{2}} \right)t_{1}^{2}} - {a_{2}t_{1}} + \frac{a_{1}}{4}} = 0},} & (13)\end{matrix}$

that can be solved analytically or numerically with a root finder. Validroots from Eq. (13) are finally used to estimate the location of thesource using Eq. (9). Note that if the sound speed is spatiallyhomogeneous, i.e. c_(i)=c₁ ∀ i, then the cubic and quartic terms vanishand the resulting quadratic equation is that found before for hyperboliclocation (Spiesberger and Wahlberg, Journal Acoustical Society ofAmerica, Vol. 112, 3046-3052, 2002).

Ambiguous solutions occur for a spatially homogeneous sound speed fieldwhen there are two positive roots to the quadratic equation. For eachambiguous source location, one can generate a model for τ₅₁ and choosethe root for t₁ that yields a model for τ₅₁ that is closest to thatmeasured. In the cases investigated here, the quartic equation (13) canyield four distinct positive values for t₁. This can only happen becausethe speed of propagation is spatially inhomogeneous. Because the valuesof c_(i) are similar here, the four distinct roots yield two pairs ofsource locations, one pair of which is relatively close to the receiversand the other pair of which is located very far from the receivers. Thedistant pair is due to the fact that locations are not exactlydetermined by intersecting hyperboloids for a spatially inhomogeneousspeed of propagation. The actual three dimensional locus of pointsspecified by a travel time difference is thus not a hyperboloid. Moreprecisely, the hyperboloid is the locus of points {right arrow over (s)}satisfying,

∥{right arrow over (r)} _(i) −{right arrow over (s)}∥−∥{right arrow over(r)} _(j) −{right arrow over (s)}∥=cτ _(ij),   (14)

where the spatially homogeneous speed of propagation is c.τ_(ij)≡t_(i)−t_(j). Letting the speed be different yields the definitionof the isodiachron which is the locus of points satisfying,

$\begin{matrix}{{{\frac{{{\overset{\rightarrow}{r}}_{i} - \overset{\rightarrow}{s}}}{c_{i}} - \frac{{{\overset{\rightarrow}{r}}_{j} - \overset{\rightarrow}{s}}}{c_{j}}} = \tau_{ij}},{c_{i} \neq c_{j}}} & (15)\end{matrix}$

which turns out to depend on the Cartesian coordinates through a fourthorder polynomial. The values of c_(i) can incorporate spatiallyinhomogeneous effects such as winds as well as wave speeds. Thecoefficients of the third and fourth powers in x, y, and z and theircombinations become very small compared to the coefficients in thesecond, first, and zero powers as the various c_(i) approach the samevalue c.

-   -   A. Geometry of Isodiachrons

The geometry of an isodiachron is described because it is a newgeometrical surface that is important in locating primary points. Amathematical definition of an isodiachron is provided in Eq. (15). Whenc_(i) and c_(j) do not depend on the coordinate of the source, the twodimensional isodiachron approximates a hyperbola in a limited region(FIG. 3). Unlike a hyperbola, for which the difference in distancebetween {right arrow over (s)} and {right arrow over (r)}_(i) and {rightarrow over (r)}_(j) and is constant, isodiachrons do not contain pointsapproaching infinity. Instead, the assumption that the difference inpropagation time be constant and that the average speeds of propagationare independent of {right arrow over (s)} constrains isodiachroniclocations to finite distances (FIG. 3).

When the speeds of propagation do not depend on the location of {rightarrow over (s)}, the Cartesian representation of the isodiachron in twodimensions is,

α₁ y ⁴+α₂ y ²+α₃=0,   (16)

where

$\begin{matrix}{\mspace{79mu} {a_{1} \equiv \frac{c_{j}^{4} - {2c_{i}^{2}c_{j}^{2}} + c_{i}^{4}}{c_{i}^{4}c_{j}^{4}}}} & (17) \\{a_{2} \equiv {2\left( {c_{i}^{4}c_{j}^{4}} \right)^{- 1}\left( {{c_{j}^{4}x^{2}} + {c_{j}^{4}x_{0}^{2}} + {c_{i}^{4}x^{2}} + {c_{i}^{4}x_{0}^{2}} - {2c_{j}^{2}x^{2}c_{i}^{2}} + {2c_{j}^{4}{xx}_{0}} - {2c_{j}^{2}c_{i}^{2}x_{0}^{2}} - {c_{j}^{4}\tau_{ij}^{2}c_{i}^{2}} - {\tau_{ij}^{2}c_{i}^{4}c_{j}^{2}} - {2c_{i}^{4}{xx}_{0}}} \right)}} & (18) \\{a_{3} \equiv {{{x^{4}\left( {c_{j}^{4} - {2c_{i}^{2}c_{j}^{2}} + c_{i}^{4}} \right)}/\left( {c_{i}^{4}c_{j}^{4}} \right)} + {\left( {c_{i}^{4}c_{j}^{4}} \right)^{- 1}\left( {{c_{j}^{4}x_{0}^{4}} + {4c_{j}^{4}x^{3}x_{0}} + {6c_{j}^{4}x^{2}x_{0}^{2}} + {4c_{j}^{4}{xx}_{0}^{3}} - {2c_{j}^{2}x_{0}^{4}c_{i}^{2}} + {\tau_{ij}^{4}c_{i}^{4}c_{j}^{4}} - {4c_{i}^{4}x^{3}x_{0}} + {6c_{i}^{4}x^{2}x_{0}^{2}} - {4c_{i}^{4}{xx}_{0}^{3}} - {2c_{j}^{4}x^{2}\tau_{ij}^{2}c_{i}^{2}} + {4c_{j}^{2}x^{2}c_{i}^{2}x_{0}^{2}} - {4c_{j}^{4}{xx}_{0}\tau_{ij}^{2}c_{i}^{2}} - {2c_{j}^{2}x_{0}^{2}\tau_{ij}^{2}c_{i}^{2}} + {c_{i}^{4}x_{0}^{4}} - {2\tau_{ij}^{2}c_{i}^{4}c_{j}^{2}x^{2}} + {4\tau_{ij}^{2}c_{i}^{4}c_{j}^{2}{xx}_{0}} - {2\tau_{ij}^{2}c_{i}^{4}c_{j}^{2}x_{0}^{2}}} \right)}}} & (19)\end{matrix}$

where the receivers are at Cartesian coordinates (−x₀, 0) and (x₀, 0).The solution for y in terms of x can be simplified by substituting z=y²in Eq. (16) which yields a quadratic equation in z. If z is real-valuedand non-negative, solutions for y are given by ±√{square root over (z)}for a given value of x. Equation (16) describes a hyperbola whenc_(i)=c_(j).

Isodiachrons take on values of y=0 at four values of x. Setting y=0 inEq. (15), the roots of x occur at (for x₀>0),

$\begin{matrix}{{x = \frac{{c_{i}c_{j}\tau_{ij}} + {\left( {c_{i} + c_{j}} \right)x_{0}}}{c_{i} - c_{j}}};{x < {- x_{0}}}} & (20) \\{{x = \frac{{c_{i}c_{j}\tau_{ij}} + {\left( {c_{i} - c_{j}} \right)x_{0}}}{c_{i} + c_{j}}};{{- x_{0}} \leq x \leq x_{0}}} & \; \\{{x = \frac{{c_{i}c_{j}\tau_{ij}} - {\left( {c_{i} + c_{j}} \right)x_{0}}}{c_{j} - c_{i}}};{x_{0} < {x.}}} & \;\end{matrix}$

These yield four roots for x for values of τ₁₂ of opposite sign.

A three-dimensional isodiachron can be formed by rotating the twodimensional isodiachron around the axis joining −x₀ to x₀ (FIG. 3). Ithas been noted above that the closed form solution for isodiachroniclocation can yield four solutions from four receivers. This can beunderstood geometrically as follows. The first pair of receiversconstrains the source to an isodiachron. A third receiver introduces asecond isodiachron which can intersect the first one along two differentellipse-like curves (e.g. in the region between x between 0.5 and 1 inpanel B of FIG. 3). A fourth receiver introduces a third isodiachronwhich can intersect the two ellipse-like curves at most four points. Inthis case, a fifth receiver is needed to determine which of the fourpoints is correct.

Because isodiachronic and hyperbolic surfaces can deviate significantlyfrom one another in the vicinity of the receivers, hyperbolic locationscan yield incorrect answers while isodiachronic locations are correct.

A related invention that is useful for applying Sequential BoundEstimation involves the analytical solution for isodiachronic locationof a primary point. Referring to FIG. 4, differences in travel time arecollected from a constellation of stations 200. Estimates for the speedof the signal are obtained between the primary point and each station210. When advection of the medium does not significantly affect thedelay of the signal between the stations and the primary point, onecomputes the location of the primary point using an analytical formula230. It is possible that the analytical equation for isodiachroniclocation does not yield a real valued solution. This can happen whenestimates of the speeds of propagation and station locations are notclose enough to their true values.

Attention now turns to FIG. 5 for a discussion of iteration. If theeffects of advection due such things as winds or currents significantlyaffect the time of travel between the primary point and a station, abound for the location of the primary point is established 240. Becausethe actual location of the primary point is unknown, the precise traveltime change between the primary point and each station due to advectionis initially unknown. So a guess is made within prior bounds for thelocation of the primary point 250. The effective speed between theprimary point and each station is computed using the vector projectionof the advection along the straight line between the primary point andeach station. The effective speed also includes the non-advectivecomponent of speed 260. For example, the speed of sound may be 330 m/s,while the effects of the winds are +1 m/s from the primary point tostation one. The effective speed from the primary point to station oneis therefore 330+1=331 m/s.

The location of the primary point is estimated using an analyticalformula given the effective speeds. If this location is close enough tothe most recent guess for this location, one exits the iterativeprocedure with a solution for the location of the primary point 280.Otherwise, the analytical solution for the location of the primary pointis used to compute another set of effective speeds to each station untilthe convergence criterion is either met 280 or exceeded 290. Inpractical application of this idea, convergence is usually obtainedafter one or two iterations of the analytical solution.

Like Sequential Bound Estimation, the algorithm shown in FIG. 4 candetect problems with the data, or assumptions about the variables byyielding no solutions for the location of the primary point 230.

In a related invention now refer to FIG. 6. In this case, the goal is toestimate any function of the probability distribution of any variableplaying a role in isodiachronic location. The word “variable” in thiscontext means either an argument to the primary function yieldinglocation or the value of this primary function.

Differences in travel time are estimated from a number of stations equalto or exceeding the number in a constellation 300. Prior bounds areestablished for all variables 310. Optionally, a database of syntheticdifferences in travel time is used to establish possibly smaller boundsfor the location of the primary point 360, 370.

If the number of stations equals the number in a constellation,probability distributions are assigned to each variable within priorbounds except for the location of the primary point that is nonlinearlyrelated to the other variables 320. Realizations of primary points areobtained by applying the analytical solution for the location of theprimary point to each set of variables using a Monte-Carlo technique330. Because this is an isodiachronic problem, the effective speed ofpropagation for one path is allowed to differ from that for another pathin the same realization of a primary point. Realizations leading toprimary points outside of its prior bounds are excluded from furtherconsideration 330. The remaining realizations are used to form anyfunction of the probability distribution of any variable 350.

Realizations of primary points here includes the possibility thatadvection can alter the travel time differences enough to represent asignificant effect. In this case, one iterates the analytical solutionuntil convergence is achieved or until convergence is not achieved, andthe realization is discarded. This idea is subsumed in FIG. 5 at step330. Details of iterating the analytical solution are described in FIG.5.

Like Sequential Bound Estimation, this algorithm can detect problemswith the data, or assumptions about the variables by yielding nosolutions for the location of the primary point. These problems stop thealgorithm and the user can investigate the cause of the problem 360,380, 330 in FIG. 6. The program can also halt execution if there arefewer than a minimum acceptable number of realizations. It must stopexecution if there are zero realizations found at step 330 or within380.

IV. Probability Density Functions for Source Location From ReceiverConstellations

When {right arrow over (r)}_(i), τ_(ij), and c_(i) are random variables,then t₁ and {right arrow over (s)} are random variables because of Eqs.(9,13). For later convenience in comparing to linear theories, considerthe situation where (r_(i)(x), r_(i)(y), r_(i)(z)), τ_(ij), and c_(i)are mutually uncorrelated Gaussian random variables with given means andvariances.

A computer generates a single random configuration of variables for aconstellation of four receivers. Each configuration consists of the set{r_(i)(x), r_(i)(y), r_(i)(z), τ_(ij), c_(i)} for some i, jε1, 2, 3, . .. ,

and i>j. Then for each constellation, of which there are a total of,

$\begin{matrix}{{{N \equiv \begin{pmatrix} \\4\end{pmatrix}} = \frac{!}{{\left( { - 4} \right)!}{4!}}},} & (21)\end{matrix}$

a source location is computed from Eq. (9) if that equation yields aunique location. If any constellation yields more than one location fromthis equation, a location is chosen to be that yielding the closestdifference in travel time to a randomly chosen fifth receiver andreceiver number one.

A valid configuration from a receiver constellation is one in which thesource location lies within some pre-determined spatial limits. Forexample, one would know that sounds from snapping shrimp occur below thesurface of the water. If a receiver constellation yields a locationabove the surface, then that particular configuration of randomvariables could not have occurred in reality, and that source locationis discarded.

Valid configurations from a receiver constellation define a cloud ofsource locations. Accurate probability density functions of locationrequire a sufficient number of valid configurations. A sufficient numberis generated for each of the N constellations.

Some constellations give better locations of the source than others.This happens for several reasons, and is usually due to the geometricalarrangement of the receivers. For example suppose the source is near thegeometric center of a pyramid and one constellation consists of fourreceivers on the vertexes of the pyramid. That constellation would beable to locate the source rather well. Then consider anotherconstellation of four receivers located close to a line at a greatdistance from the source. This constellation would not be able to locatethe source as well.

The only physically possible locations for the source lie within theintersections of the clouds. The other source locations are invalid.Here, the upper and lower bounds of the intersected region, ({circumflexover (X)}, {hacek over (X)}), (Ŷ, {hacek over (Y)}), and ({circumflexover (Z)}, {hacek over (Z)}), are estimated along the Cartesian axes. If({circumflex over (x)}_(c), {hacek over (x)}_(c)), (ŷ_(c), {hacek over(y)}_(c)), and ({circumflex over (z)}_(c), {hacek over (z)}_(c)) denotethe maximum and minimum values of x, y, and z for cloud c, then theregion of intersection is,

{circumflex over (X)}=min({circumflex over (x)} ₁,{circumflex over (x)}₂ ,{circumflex over (x)} ₃ , . . . {circumflex over (x)} _(N))

{hacek over (X)}=max({hacek over (x)} ₁,{hacek over (x)} ₂ ,{circumflexover (x)} ₃ , . . . {circumflex over (x)} _(N))

Ŷ=min(ŷ ₁ ,ŷ ₂ ,ŷ ₃ , . . . ŷ _(N))

{hacek over (Y)}=max({hacek over (y)} ₁ ,{hacek over (y)} ₂ ,{hacek over(y)} ₃ , . . . {hacek over (y)} _(N))

{circumflex over (Z)}=min({circumflex over (z)} ₁ ,{circumflex over (z)}₂ ,{circumflex over (z)} ₃ , . . . {circumflex over (z)} _(N))

{hacek over (Z)}=min({hacek over (z)} ₁ ,{hacek over (z)} ₂ ,{hacek over(z)} ₃ , . . . {hacek over (z)} _(N))   (22)

For each cloud c, locations outside of the bounds in Eq. (22) arediscarded. New probability density functions are formed from theremaining locations in each cloud for the x, y, and z values separately.It is not expected that the new probability density functions wouldasymptotically approach one another because each constellation will beable to locate the source with a different quality.

The probability density functions for the source in its x, y, and zcoordinates are dependent distributions. This can be accommodated byforming the joint probability density function of location for eachcloud if desired. This is not done here.

P percent confidence limits are estimated by finding the P percentconfidence limits for each cloud separately using its probabilitydensity functions in x, y, and z. The Cartesian upper and lower boundsfor a specified confidence limit for constellation c are denoted[{circumflex over (P)}_(c)(x), {hacek over (P)}_(c)(x)], [{circumflexover (P)}_(c)(y), {hacek over (P)}_(c)(y)], and [{circumflex over(P)}_(c)(z), {hacek over (P)}_(c)(z)]. The final bounds for the sourceare chosen from the smallest bound for x, y, and z from eachconstellation. For example, suppose constellation p has the smallestvalue of {circumflex over (P)}(x)−{hacek over (P)}(x) for all values ofc, constellation q has the smallest value of {circumflex over(P)}(y)−{hacek over (P)}(y) for all values of c, and constellation r hasthe smallest value of {circumflex over (P)}(z)−{hacek over (P)}(z) forall values of c. Then the final confidence limits for the source are[{circumflex over (P)}_(p)(x), {hacek over (P)}_(p)(x)], [{circumflexover (P)}_(q)(y), {hacek over (P)}_(q)(y)], and [{circumflex over(P)}_(r)(z), {hacek over (P)}_(r)(z)].

V. Examples of Hyperbolic and Isodiachronic Locations

Examples below utilize 2000 valid configurations of random variables toestimate probability density functions for each cloud.

-   -   A. Hyperbolic Location

Because error bars derived from linearized hyperbolic locationtechniques assume the signal speed is spatially homogeneous, comparisonwith a linear theory is done using a spatially homogeneous value of 1475m/s. The travel time differences are computed for this speed. In theerror analysis, the sound speed is assumed to be a Gaussian randomvariable with mean 1475 m/s and standard deviation of 10 m/s. Thisstandard deviation is realistic if one considers paths emitted from ashallow source to receivers at perhaps 3 m and 100 m depth because thesurface region can be very warm compared with temperatures below. Traveltime differences are assumed to be mutually uncorrelated Gaussian randomvariables with means given by true values and standard deviations of0.0001414 s. Receiver locations are assumed to be mutually uncorrelatedGaussian random variables with means given by their true values andstandard deviations as shown in Table I.

TABLE I x (m) y (m) z (m) ARRAY 1 R1 0 ± 0 0 ± 0 0 ± 0 R2 1000 ± 2   0 ±0 0 ± 1 R3 1000 ± 2   1000 ± 2   0 ± 1 R4 0 ± 2 1000 ± 2   0 ± 1 R5 0 ±5 0 ± 5 −100 ± 5    SOURCE 551 451 −100 ARRAY 2 R1 0 ± 0 0 ± 0 0 ± 0 R21414 ± 2   0 ± 0 0 ± 1 R3 534 ± 2  400 ± 2  0 ± 1 R4 1459 ± 5   −1052 ±5     −25 ± 5    R5 1459 ± 20  −1052 ± 20    −95 ∓ 20   R6  0 ± 20  0 ±20 −100 ± 20    R7 1414 ± 20   0 ± 20 −100 ± 20    SOURCE 860 47 −5

The errors in the travel time differences and receiver locations may notbe mutually uncorrelated as assumed above. For example, errors in τ_(i1)may be correlated with τ_(j1) because both involve t₁. Also, forexample, the errors in the location of receiver 3 are not necessarilyuncorrelated between two constellations that both contain receiver 3.Linear error analysis can accommodate correlations between randomvariables, as can the non-linear analysis done here. However,incorporating these correlations leads to larger computational times inthe non-linear analysis, so they are not implemented. Error models thatfollow are done assuming that random variables are mutually uncorrelatedfor both the linear and non-linear analysis. We found that the resultswere identical for the linear analysis (Spiesberger and Wahlberg, 2002)when the random variables were correlated.

To mimic marine examples, array one has five receivers separated byO(1000) m horizontally and up to 100 m vertically (FIG. 7, Table I). Thefive receiver constellations (Eq. 21), one through five, are {1, 2, 3,4}, {1, 2, 3, 5}, {1, 2, 4, 5}, {1, 3, 4, 5}, and {2, 3, 4, 5}respectively. The 100% confidence limits for the source are computedfrom Eq. (22) for increasing numbers of constellations where N is set to1, 2, 3, 4, and 5 respectively in this equation for FIG. 8. As moreclouds are intersected, the limits for the source decreasemonotonically, with the biggest improvement occurring with the additionof constellation two with one. Constellation one only uses the receiversat z equal zero. Constellation two is the first one that includes thereceiver at z=−100 m. This deeper receiver not only helps in locatingthe source's vertical coordinate, but significantly helps locate thehorizontal coordinates as well. The probability density functions forthe source location come from constellations two and four (FIG. 9).These density functions appear to be approximately Gaussian. The 68%confidence limits span only a few meters in the horizontal coordinatesand are about 50 m in the vertical coordinate (Table II). The confidencelimits from the non-linear analysis are about a factor of ten less thanthose from the standard linear analysis of errors in the horizontalcoordinates. The non-linear analysis has smaller limits because thehyperboloids are not well approximated by planes in the horizontaldirections as required by the linear analysis. Non-linear analysisyields somewhat larger limits in z than linear analysis (Table II).Similarity of results in z indicates that the hyperboloids are fairlywell approximated as planes in the vertical coordinate in the vicinityof the source.

TABLE II CARTESIAN 68% CONFIDENCE LIMITS (m) COORDINATE NON-LINEARLINEAR ARRAY 1 x 550 to 552 530 to 573 y 450 to 452 432 to 471 z −145 to−77  −121 to −80  ARRAY 2 x 860 to 865  236 to 1486 y 46 to 62 −1890 to1988   z −138 to 65    −163 to 153  

Array two has seven receivers with the largest horizontal and verticalseparations being about 1400 and 100 m respectively (FIG. 7). There are35 receiver constellations (Eq. 21) of which the first four provide mostof the accuracy for locating the source at the 100% confidence limits(FIG. 10). These constellations are the first of the 35 that include allthe deeper receivers. There are modest increases in accuracy from otherconstellations, most notably 29 and 33. The probability densityfunctions in the x-y-z coordinates come from constellations 7, 4, and 26respectively (FIG. 11). The distributions in x and y do not look veryGaussian, while the distribution in z looks more Gaussian-like. Thesedepartures from Gaussian distributions are quite different than theGaussian distributions usually assumed from linear analysis. This time,the 68% confidence limits from the non-linear analyses are two orders ofmagnitude smaller than those from standard linear analysis in x and y(Table II). The linear and non-linear confidence limits are similar forthe vertical coordinate.

-   -   B. Isodiachronic Location

Consider a situation where the speed of sound is similar, but notexactly the same between each source and receiver. Consider anatmospheric example for locating a sound at Cartesian coordinate(20,100,7) m from five receivers at (0,0,0), (25,0,3), (50,3,5),(30,40,9), and (1,30,4) m respectively. The speed of sound is a typical330 m/s. The speed of propagation is made to be inhomogeneous byintroducing a wind of 10 m/s in the positive y direction. Next,simulated values of the travel time differences are computed using thesevalues. The source is located using hyperbolic and isodiachroniclocation. It will be seen that only isodiachronic location yields acorrect solution.

All Gaussian random variables in this simulation are truncated to have amaximum of two standard deviations for the following two reasons. First,many experimental situations are inaccurately represented by assumingthat random variations differ from an estimate by say 10 standarddeviations. Instead, it is more realistic to truncate the variations.Second, it is important to note that a realistic truncation is easy toimpose with the models developed here but is difficult to implement withanalytical and linear approximations for error.

The standard deviation for receiver locations is 0.02 m. The variationsare zero for the x, y, and z coordinates of receiver 1, the y and zcoordinates of receiver 2, and the z coordinate of receiver 3. Thecoordinates with zero variations merely define the origin andorientation of the coordinate system. The x and z components of thewinds are modeled to have a value of 0 m/s. For hyperbolic location, thespeed of acoustic propagation must be spatially homogeneous. The meanand standard deviation for sound speed are 330 m/s and 10 m/srespectively. For isodiachronic location, the speed of propagation isinhomogeneous. The speed of sound is taken to be 330 m/s. The a priorivalue of the wind in the y direction has mean 0 and standard deviation10 m/s. Values of the speed of acoustic propagation, c_(i) betweenreceiver i and the source are unknown because the location of the sourceis initially unknown. Therefore, it is impossible to precompute thecomponent of the wind vector along the direction from the source to eachreceiver. Instead, the value for each c_(i) is computed using aniterative procedure. First, the spatial limits for the source areobtained by searching a pre-computed data base of travel timedifferences as a function of source location. These travel timedifferences are generated using the assumptions for receiver,measurement, and environmental uncertainties described above. The firstguess for the location of the source is obtained by choosing a locationat random within the spatial bounds. Next, the effects of winds on thedata are computed by using this location guess and the locations of thereceivers into account. Then the speed of propagation between the sourceand each receiver is computed, and a solution for source location isobtained from the quartic equation for isodiachronic location. Thissolution is used as a new guess for the location of the source, fromwhich a new set of speeds of propagation are computed between thissource and the receivers, and again an isodiachronic solution for sourcelocation is obtained. This procedure continues until the differencebetween the guessed location and the location from the isodiachroniclocations reaches a specified small enough threshold, at which pointiteration ceases. The only points used to form points within the cloudsare those for which the iteration converges.

The error in travel time due to the straight path approximation istypically less than a microsecond at these ranges. The travel timedifferences are derived with ideal values for means and from a standarddeviation of 16 μs. The 16 μs value is derived from standard methods foran r.m.s. bandwidth of 1000 Hz and a peak signal-to-noise ratio of 20 dBin the cross-correlation function of the signals between receivers.

Incorrect locations are obtained using the hyperbolic method. Forexample, the source's 100% confidence limits for y are 102.0 to 108.1 mm, but its actual y coordinate is 100 m. So given a priori variations ofreceiver locations, travel time differences, and environmentalvariations, the hyperbolic method always yields incorrect answers.

With isodiachronic location, 95% confidence limits for the source are x:16.6 to 20.5 m, y: 98.8 to 101.7 m, z: 3.5 to 40.5 m. These arestatistically consistent with the correct answer. Other confidencelimits could be given but they are not shown because the point is thatisodiachronic location yields a correct answer at a stringent confidenceof 95%.

Another case where isodiachronic location would be useful is one wherethe speed of sound is quite different between the source and eachreceiver. In this case, hyperbolic locations would be inappropriate touse because the speed of sound is not nearly constant in space. Forexample suppose low frequency sources such as Finback whales arelocated.

Suppose some receivers close to the source pick up only the firstacoustic path through the sea, while other distant receivers pick uponly the acoustic path that propagates below the sea-floor because thepaths through the water are blocked by the sea-mounts. The speed ofpropagation along the water and solid-Earth paths can differ by morethan a factor of two (Spiesberger and Wahlberg, 2002). In otherscientific fields, sounds can propagate to receivers along paths withdifferent speeds of sound, such as from vehicles where paths propagatethrough the air and ground.

-   -   C. Experimental Data        -   1. First Experiment

An experimental example is given for hyperbolic and isodiachroniclocations.

On 24 Jun. 1991, eight microphones were set up at the inventor's houseat 15 Redwing Terrace, North Falmouth, Mass. Signals from themicrophones were cabled to a PC were the signals were digitized.Acoustic signals were played from an audio speaker at Cartesiancoordinate (11.07±0.05, 14.67±0.05, −4.82±0.13) m. The coordinates ofthe eight microphones were (0,0,0), (20.064, 0, −4.947), (32.876,13.637, −4.649), (17.708, 22.766, −4.479), (1.831, 15.718, 0.951),(19.888, 9.177, −4.570), (13.710, 6.437, −0.518), and (28.982, 18.491,−6.515) m. The locations were measured optically and were accuratewithin ±0.1 m horizontally and ±0.026 m vertically. The relativehumidity was about 55% and the temperature was about 23.5° C., given aspeed of sound of about 347 m s⁻¹. The speed of the wind was less than 1m s⁻¹. The speaker emitted a narrowband signal near 1700 Hz with abandwidth of a few hundred Hertz.

The accuracy of the straight path approximation is about 0.1 μs forthese conditions. The signals from microphones one through eight arecross-correlated, yielding estimates for τ_(j1), j=2, 3, 4, . . . 8.

For hyperbolic location, the speed of sound is allowed to vary by ±4 ms⁻¹. Each Cartesian components of the wind varies up to ±2 m s⁻¹. Thereare 35 constellations of receivers with eight receivers, where only theconstellations involving τ_(j1), j=2, 3, 4, . . . 8 are used. The 100%limits for the source are for x: 10.94 to 11.17 m; y: 14.58 to 14.88 m;z: −5.26 to −4.16 m. These are consistent with the optically surveyedlocation of the source.

For isodiachronic location, the same values are assumed for the randomvariables. The data base of travel time differences yields bounds forthe source within x: 10 to 14 m; y: 12 to 16 m; z: −10 to +6 m. Withinthese bounds the isodiachronic location method yields 100% limits forthe source within x: 10.79 to 11.42; y: 14.46 to 15.00; z: −5.48 to−3.65 m. These are consistent with the location of the speaker.

-   -   2. Second Experiment

On 4 Jun. 1995, five omni-directional microphones recorded sounds from aRedwing Blackbird, in Port Matilda, Pa. (Bird B1 in FIG. 3 ofSpiesberger, Journal Acoustical Society America, Vol. 106, 837-846,1999). Isodiachronic location is done using published data (Table 2 inSpiesberger, 1999). The reference speed of sound is 344 m/s. A prioridistributions of other variables are taken to be Gaussian and truncatedat one standard deviation. The standard deviations for the differencesin travel times is 0.000067 s. The standard deviation of the speed ofsound is 2 m/s. The standard deviation of each horizontal component ofthe wind is 2 m/s. The standard deviation of the vertical component ofthe wind is 0.5 m/s. The standard deviation of the travel time due tothe straight path approximation is 0.1 μs. The standard deviation ofeach Cartesian coordinate of a receiver is 0.05 m/s except as follows.

Receiver one is defined to be at the origin without error. Receiver twois defined such that the y axis passes through it and its z coordinateis zero. Receiver three is defined such that its z coordinate is zero.These definitions merely define the absolute location and rotation ofthe coordinate system (Table III).

TABLE III CARTESIAN COORDINATE (m) RECEIVER x y z R1 0 0 0 R2 19.76 0 0R3 17.96 −18.80 0 R4 2.34 −29.92 −0.02 R5 −12.41 −14.35 −0.43

With five receivers, there are five constellations of receivers.Hyperbolic locations and their probability distributions are estimatedusing Sequential Bound Estimation. A priori bounds for the location ofthe Redwing Blackbird are taken to be huge, namely x and y: −10⁹ to 10⁹m; z: −2900 m to +100 m. Cartesian coordinates have 95% and 100%confidence limits of (9.74±0.038, 7.79±0.50, 0.36±4.8) m and(9.74±0.048, 7.68±0.73, 0.43±5.3) m respectively. Both estimates areconsistent with 95% confidence limits for the location of the bird,namely (9.8±0.5, 6.8±0.5, 2.3±1) m (Table IV in Spiesberger, 1999).

With isodiachronic location, a priori distributions of random variablesare the same except the speed of sound is allowed to independently varyfor each section with standard deviation of 2 m/s, truncated at onestandard deviation as before. The a priori bounds for the location ofthe Redwing Blackbird are smaller than before, namely x: 5 to 10 m; y: 5to 15 m; z: −5 to +10 m. These limits are obtained using a simulateddatabase as will be explained shortly. Cartesian coordinates have 95%and 100% confidence limits of (9.73±0.15, 7.74±1.3, 0.58±5.4) m and(9.73±0.22, 7.96+1.9, 0.98±6.0) m respectively. These are consistentwith the location of the bird (above).

The database of simulated data is constructed by subdividing Cartesianspace within x and y: −30 to +30 m and z from −5 to +10 m with cells ofrectangular parallelepipeds with sides of length about 3 m. Within eachcell, the probability distribution of a simulated bird is assumed to beuniformly distributed. Probability distributions for the speeds ofpropagation, winds, and locations of the receivers are the same as usedfor isodiachronic location as described above. Monte-Carlo realizationsare used to generate thousands of differences in travel times from eachcell. The maximum and minimum values of the simulated differences intravel time are stored in a database. The data of travel timedifferences are compared to the simulated maximum and minimum valuesfrom each cell. If the maximum and minimum values of all simulatedtravel time differences are consistent with measured travel timedifferences within measurement error, the cell is a possible locationfor the Redwing Blackbird. Otherwise that cell is not further consideredfor the location of the bird. After applying this method, the a prioribounds for the Redwing Blackbird reduce to x: 5 to 10 m; y: 5 to 15 m;z: −5 to +10 m.

-   -   D. Global Positioning System

The Global Positioning System (GPS) allows receivers to locatethemselves through measurements of the difference in travel time ofelectromagnetic waves between synchronized emissions from the satellitesand the receiver. Location is traditionally achieved using hyperboliclocation techniques.

-   -   -   1. Standard Error Model L1 C/A and No Selective Availability

The simulation of location accuracy with isodiachronic location is madefor a GPS receiver using the Standard Error Model L1 C/A and noSelective Availability (SA) (Table IV). Using the errors in Table IV, acommercially available GPS receiver can locate itself horizontallywithin 10.2 m at the 68% confidence interval (Table 2 inhttp://www.edu-observatory.org/gps/check_accuracy.html). Using identicalerrors, isodiachronic location yields a 68% confidence limit of 6.3 m,about half that obtained from a standard GPS receiver.

The isodiachronic location assumes that location is made from only oneset of data from eight GPS receivers (Table V), so there is no averagingor filtering in time or space of any data. The increase in locationaccuracy is solely due to the way that location is estimatedisodiachronically. The isodiachronic location uses no additionalinformation over and above that collected by a GPS receiver.

TABLE IV Type of Error 68% Confidence limit Satellite position for eachof 2.1 m (x, y, z) Cartesian coordinates Satellite clock bias  2. m ofpath distance Satellite clock random error 0.7 m Ionosphere bias   4 mTroposphere bias 0.5 m Multipath bias   1 m Receiver measurement bias0.5 m Receiver random measurement 0.5 m error

TABLE V Satellite Elevation (deg) Azimuth (deg) GPS BIIA-23 (PRN 04)40.1 222.9 GPS BIIA-20 (PRN 07) 65.2 319.5 GPS BIIA-21 (PRN 09) 14.7312.8 GPS BIIR-03 (PRN 11) 18.67 51.8 GPS BIIR-04 (PRN 20) 37.0 87.3 GPSBII-08 (PRN 21) 36.4 84.7 GPS BIIA-11 (PRN 24) 8.2 223.3 GPS BIIR-05(PRN 28) 64.2 149.7

In the simulation above, Sequential Bound Estimation is used tosimultaneously solve for the location of the GPS receiver, the speed oflight between the receiver and each satellite, and for the correction oflocation for each satellite.

-   -   -   2. Precise Error Model, Dual-frequency, P(Y) Code, No            Selective Availability, No Differential Capability

This simulation is similar to that in the previous section except thisis for a dual-frequency GPS unit without access to differential data.

The simulation of location accuracy with Sequential Bound Estimation ismade for a Precise error model, dual-frequency, P(Y) code with errorsgiven by Table 4 ofhttp://www.edu-observatory.org/gps/check_accuracy.html. The errors fromthis table are repeated in Table VI here. This case does not haveselective availability.

Using the errors in Table VI, a commercially available GPS receiver canlocate itself horizontally within 6.6 m at the 68% confidence interval(Table 4 in http://www.edu-observatory.org/gps/check_accuracy.html).Using identical errors, isodiachronic location yields a 68% confidencelimit of 4.3 m.

The isodiachronic location is made from only one set of data from eightGPS receivers (Table V), so there is no averaging or filtering in timeor space of any data. It appears that the increase in location accuracyis solely due to the way that location is estimated. The isodiachronicmethod uses no additional information over and above that collected by aGPS receiver.

TABLE VI Type of Error 68% Confidence limit Satellite position for eachof 2.1 m (x, y, z) Cartesian coordinates Satellite clock bias  2. m ofpath distance Satellite clock random error 0.7 m Ionosphere bias   1 mTroposphere bias 0.5 m Multipath bias   1 m Receiver measurement bias0.5 m Receiver random measurement 0.5 m error

In this simulation, Sequential Bound Estimation is used tosimultaneously solve for the location of the GPS receiver, the speed oflight between the receiver and each satellite, and for the correction oflocation for each satellite.

-   -   -   3. Differential GPS Location With Dual Frequency

This simulation is similar to that in the previous section except thisis for a dual-frequency GPS unit with access to differential data from areference station.

The simulation of location accuracy with Sequential Bound Estimation ismade for a Precise error model, dual-frequency, P(Y) code with errorsgiven by Table 4 ofhttp://www.edu-observatory.org/gps/check_accuracy.html for differentialGPS. The errors from this table are repeated in Table VII here. Thiscase does not have selective availability.

Using the errors in Table VII, a commercially available GPS receiver canlocate itself horizontally within 3 m at the 68% confidence interval(Table 4 in http://www.edu-observatory.org/gps/check_accuracy.html).Using identical errors, isodiachronic location yields a 68% confidencelimit of 1.7 m.

The isodiachronic location assumes that location is made from only oneset of data from eight GPS receivers (Table V), augmented with data froma differential GPS station, so there is no averaging or filtering intime or space of any data. It appears that the increase in locationaccuracy is solely due to the way that location is estimatedisodiachronically. The isodiachronic algorithm uses no additionalinformation over and above that collected by a GPS receiver.

TABLE VII Type of Error 68% Confidence limit Satellite position for eachof 2.1 m (x, y, z) Cartesian coordinates Satellite clock bias  0. m ofpath distance Satellite clock random error  0. m Ionosphere bias 0.1 mTroposphere bias 0.1 m Multipath bias 1.4 m Receiver measurement bias0.5 m Receiver random measurement  0. m error

In this simulation, Sequential Bound Estimation is used tosimultaneously solve for the location of the GPS receiver, the speed oflight between the receiver and each satellite, and for the correction oflocation for each satellite.

VI. Sequential Bound Estimation

It is useful to further discuss Sequential Bound Estimation to exemplifyits workings.

For hyperbolic and isodiachronic locations in three spatial dimensions,steps for Sequential Bound Estimation could be,

-   -   1. Partition the stations into groups of four. For R stations        there are P=R!/(R−4)!4! such groups called “constellations.”. If        a fifth station is needed to resolve a solution ambiguity, that        fifth station is chosen at random from the remaining R−4        stations.    -   2. Form a priori distributions for variables affecting the        location of the primary point. Those variables are the locations        of the stations, the speeds of signal propagation, the winds or        currents (if any), and the travel time differences.    -   3. Use Monte-Carlo simulations to generate spatial clouds for        the location of the primary point for each of the P groups.    -   4. Form upper and lower bounds for the primary point by        intersecting the P clouds. This can be done at the end or        sequentially.    -   5. Discard primary points for each group that are outside the        intersected region.    -   6. Form new probability distributions for the primary point for        each group from the remaining primary points within the        intersected region.    -   7. The probability distribution for the primary point location        is chosen from the group giving the smallest specified        confidence limits for each of the x, y, and z Cartesian        coordinates.        -   1. Incorporating Temporal Evolution from Models

Like a Kalman filter, Sequential Bound Estimation can incorporate amodel to transition variables from one time to another. A simple modelis used here to demonstrate how Sequential Bound Estimation is used toestimate locations of receivers and the sound speed field from atemporal sequence of sounds at unknown locations. This jointlocation/tomography problem is nonlinear. It is not known if a correctsolution can be achieved by linearizing the equations that connect thesevariables.

The bounds for the temporal evolution from times t₁ to t₂ of the soundspeed field, relative to a reference speed, are given by,

$\begin{matrix}{{{\delta {{\hat{c}}_{r}\left( t_{2} \right)}} = {\min \left\{ {{+ {c_{\max}}},{{{\hat{c}}_{r}\left( t_{1} \right)} + {{{t_{2} - t_{1}}}{\frac{{\delta}\; c_{r}}{t}}}}} \right\}}},} & (23) \\{{{\delta {{\overset{ˇ}{c}}_{r}\left( t_{2} \right)}} = {\max \left\{ {{- {c_{\max}}},{{{\overset{ˇ}{c}}_{r}\left( t_{1} \right)} - {{{t_{2} - t_{1}}}{\frac{{\delta}\; c_{r}}{t}}}}} \right\}}},} & \;\end{matrix}$

where the average deviation of speed between the source and receiver ris δc_(r)(t), the maximum deviation is |c_(max)|, the maximum allowedrate of change of the fluctuation is

${\frac{{\delta}\; c_{r}}{t}},$

and upper and lower bounds are denoted by ̂ and {hacek over ( )}respectively. Eq. (23) is a method for relaxing prior constraints onbounds to maximum limits at a specified maximum rate of change. In theabsence of further data, the bounds expand to a priori values. Boundsfor the winds obey similar relationships.

In the simulations given later, it is assumed that receivers do notmove, but their locations are uncertain. Thus the spatial limits forreceiver r for group p are those determined from the previous group'slimits; i.e. the limits determined from group p−1.

-   -   B. Surveying with Data

Sequential Bound Estimation is applied to provide information about thelocation of the x coordinate of receiver two (Table III) from Sec. V,part C, No. 2 above. The information is provided only from the soundsrecorded on the five microphones from one call of a Redwing Blackbird.In particular, the a priori statistics are unchanged except as follows.The reference location of receiver two is changed from (19.76, 0, 0) to(21, 0, 0). The a priori probability distribution is Gaussian with meanof 21 m and standard deviation of 1.5 m, with variation truncated at onestandard deviation. It is important to note that the algorithm does notknow that the x coordinate of receiver two is really 19.76±0.05 m butinstead only knows that it is 21±1.5 m.

Sequential Bound Estimation is applied to each of the five receiverconstellations in sequential order. At the end of this algorithm, it isfound that the 95% confidence limits for the x coordinate of receivertwo are between 19.50 and 21.98 m. This is consistent with the correctanswer of 19.76±0.05 m. Additionally, the 95% confidence interval 19.5to 21.98 m, is smaller than the a priori range from 19.5 to 22.5 m.Furthermore, the resulting probability distribution of the x coordinateis highly skewed toward 19.76 m (FIG. 12). Therefore, Sequential BoundEstimation yields statistically significant information about thelocation of the x coordinate of receiver two that is obtained onlythrough use of the sounds from a single call of a Redwing Blackbird.

Simulation for Estimation of Signal Propagation Speed

Sequential bound estimation is used to find a probability distributionfor the homogeneous component of the sound speed when the locations ofthe receivers are well known. Six receivers are distributed in air overa region of about 30 m by 30 m (FIG. 13). Their elevations are about 2 mexcept for receiver 6 which is at an elevation of 7 m. The initiallyunknown locations for 10 sources are chosen at random (FIG. 13). Thesources are assumed to produce sound at 10 s intervals. The elevationsof the sources are assumed to be between 3 and 6 m.

The errors in the travel time differences are taken to be Gaussian withstandard deviation of 0.0001 s, but the errors are truncated at onestandard deviation.

The speed of sound is composed of spatially homogeneous andinhomogeneous components.

The homogeneous component is about 329.5 m s⁻¹. It is allowed to vary ata maximum rate of 0.01 m s⁻², so in 10 s, it can vary by 0.1 m s⁻¹.

The initial estimates for the receiver locations are incorrect. TheCartesian coordinate of each receiver is selected from a uniformdistribution having a span of ±0.05 m on either side of the true value.No error is assigned to the location of receiver one (defined to be theorigin), to the y and z coordinates of receiver two, and the zcoordinate of receiver three because these merely define the origin andorientation of the coordinate system. The a priori errors for theinitial estimates are taken to be independently distributed Gaussianrandom variables with means at the incorrect locations and standarddeviations of 0.098 m, except the distributions are truncated at onestandard deviation.

Mild values for the winds are assumed. The mean values for the winds inthe x, y, and z directions are zero. The x, y, and z components of thewinds are allowed to fluctuate as Gaussian random variables with meanzero and standard deviations of 1 m s⁻¹ except for the z component witha standard deviation of 0.5 m s⁻¹. The variations are truncated at onestandard deviation. The maximum rate of change of any component of thewind is 1 m s⁻².

The speed of sound is composed of spatially homogeneous andinhomogeneous components. The inhomogeneous component may vary fromsection to section by up to ±0.1 m s⁻¹ at a maximum rate of 0.01 m s⁻².The actual value for the homogeneous component is about 329.5 m s⁻¹. Itis allowed to vary at a maximum rate of 0.01 m s⁻², so in 10 s, it canvary by 0.1 m s⁻¹. Before data are taken, it is assumed that thespatially homogeneous component of the sound speed is uniformlydistributed in the interval 330±50 m s⁻¹. Following the use of sourceone, the lower and upper bounds for the deviation are 330−6.9 and330+5.3 m s⁻¹. Following the use of source two, these bounds are 330−3.1and 330+2.7 m s⁻¹. These bounds do not change significantly followingthe use of sources three through ten. The correct value of the speed ofsound, about 329.5 m s⁻¹, falls within the bounds provided by SequentialBound Estimation. Sequential Bound Estimation shows the initialuncertainty of ±50 is reduced by a factor of about 16. An uncertainty of±3 m s⁻¹ is equivalent to ±5° C.

Consider adding four additional receivers at Cartesian coordinates (20,0, 3), (−20, 22, 4), (25, 21, 2), and (−5, 3, 3) m. Modeling theenvironment and receiver errors in the same way as above, it is foundthat the homogeneous component of the sound speed field has 100%confidence limits between 330−2.5 m s⁻¹ and 330+1.7 m s⁻¹ following theuse of the first three sources. The uncertainty of about ±2.1 m s⁻¹ isequivalent to ±3.5° C. The source locations are different than above,but are similarly situated and not chosen to optimize any measure ofperformance.

VII. Additional Approaches for Location

-   -   A. Constraints

Besides the use of travel time differences themselves, locations may beconstrained by other data or equations. For example, it may be knownthat an emitter of sound or electromagnetic energy is on the surface ofthe Earth. There may be an equation describing this surface. Thisconstraint reduces the number of receivers required to obtain anunambiguous solution.

It is important to distinguish between an ambiguous solution and theinfinite number of location solutions arising from errors in variables.The distinction can be made by assuming for the moment that measurementsof arrival time difference are made without error and the variables suchas the speeds of signal propagation are known exactly and the locationsof stations are known exactly. Then, if the equations for location yielda unique solution, an unambiguous solution exists. Otherwise, anambiguous solution exists. Schmidt (1972) identifies regions whereambiguous and unambiguous solutions occur for hyperbolic location. Whenone adds errors in measurements and uncertainties to the variables, theactual location of the primary point becomes blurred into a region ofspace containing an infinite number of locations in this bounded region.

The addition of a constraint on the location of a primary point cansometimes reduce the number of stations required to obtain anunambiguous solution.

One can use many types of constraints to reduce the number of receiversin a constellation. Constraints that have been used include differentialDoppler (when there is relative motion between the primary point and thestations), altitude measurements of a primary point being located withGPS, a priori information about the location of a primary point, etc.

VIII. Ellipsoidal and Isosigmachronic Location: Theory

-   -   A. Isosigmachronic Location

When the speed of propagation between one station and a primary pointdiffers from the speed between the primary point and a different stationlocation, ellipsoidal location can no longer be utilized in a strictsense. The method for obtaining an analytical solution for the locationof a primary point in this situation is derived here in a manneranalogous to that used for isodiachronic location. Instead of an ellipsedefined by the locus of points along which the sum of distances isconstant from two stations, an “isosigmachron” is defined to be thelocus of points along which the sum of travel times is constant from twostations when the speeds differ.

-   -   -   1. Solution Using Four Receiver Constellation

Let the location of a primary point be P≡[x y z]^(T) in Cartesiancoordinates where T denotes transpose. Suppose a signal from a source at{right arrow over (s)} propagates to the primary point with effectivespeed c₀ with a time of t₀ (FIG. 14). Then suppose the signal thenpropagates to station r_(i) from primary point {right arrow over (P)}with effective speed c_(i) and time t_(i). It is assumed that t₀+t_(i)is known unlike the situation for hyperbolic or isodiachronic locationin which only differences in arrival time are known at the outset.

Define the sum of travel times as,

T _(i) ≡t ₀ +t _(i),   (24)

so

t _(i) =T _(i) −t ₀.   (25)

The distance squared between {right arrow over (P)} and {right arrowover (r)}_(i) is,

∥{right arrow over (P)}−{right arrow over (r)} _(i)∥² =c _(i) ² t _(i)².   (26)

Put Eq. (25) in (26) to get,

∥{right arrow over (P)}−{right arrow over (r)} _(i)∥² =c _(i) ²(T _(i)−t ₀)².   (27)

Without loss of generality, the origin is defined to be at {right arrowover (r)}₁ so {right arrow over (r)}₁={right arrow over (0)}. For i=1 inEq. (27) we get,

∥{right arrow over (P)}∥ ² =c ₁ ²(T ₁ −t ₀)².   (28)

Subtract Eq. (28) from (27) to get,

∥{right arrow over (P)}−{right arrow over (r)} _(i)∥² −∥{right arrowover (P)}∥ ² =c _(i) ²(T _(i) −t ₀)² −c ₁ ²(T ₁ −t ₀)², , i=2, 3, 4

which simplifies to,

$\begin{matrix}{{{R\overset{\rightarrow}{P}} = {\frac{{\overset{\rightarrow}{b}}_{e}}{2} - {t_{0}{\overset{\rightarrow}{f}}_{e}} - {t_{0}^{2}{\overset{\rightarrow}{g}}_{e}}}},{i = 2},3,4} & (29)\end{matrix}$

where

$\begin{matrix}{{R \equiv \begin{pmatrix}{r_{2}(x)} & {r_{2}(y)} & {r_{2}(z)} \\{r_{3}(x)} & {r_{3}(y)} & {r_{3}(z)} \\{r_{4}(x)} & {r_{4}(y)} & {r_{4}(z)}\end{pmatrix}};} & (30)\end{matrix}$

When the inverse of R exists, which is the usual case, Eq. (29) isequivalent to,

$\begin{matrix}{{\overset{\rightarrow}{P} = {{R^{- 1}\frac{{\overset{\rightarrow}{b}}_{e}}{2}} - {t_{0}R^{- 1}{\overset{\rightarrow}{f}}_{e}} - {t_{0}^{2}R^{- 1}{\overset{\rightarrow}{g}}_{e}}}},{i = 2},3,4} & (31)\end{matrix}$

where,

$\begin{matrix}{{{\overset{\rightarrow}{b}}_{e} \equiv \begin{pmatrix}{{{\overset{\rightarrow}{r}}_{2}}^{2} + \left( {{c_{1}^{2}T_{1}^{2}} - {c_{2}^{2}T_{2}^{2}}} \right)} \\{{{\overset{\rightarrow}{r}}_{3}}^{2} + \left( {{c_{1}^{2}T_{1}^{2}} - {c_{3}^{2}T_{3}^{2}}} \right)} \\{{{\overset{\rightarrow}{r}}_{4}}^{2} + \left( {{c_{1}^{2}T_{1}^{2}} - {c_{4}^{2}T_{4}^{2}}} \right)}\end{pmatrix}};} & (32) \\{{{{\overset{\rightarrow}{f}}_{e} \equiv \begin{pmatrix}{{T_{1}c_{1}^{2}} - {T_{2}c_{2}^{2}}} \\{{T_{1}c_{1}^{2}} - {T_{3}c_{3}^{2}}} \\{{T_{1}c_{1}^{2}} - {T_{4}c_{4}^{2}}}\end{pmatrix}};{{\overset{\rightarrow}{g}}_{e} \equiv {\frac{1}{2}\begin{pmatrix}{c_{2}^{2} - c_{1}^{2}} \\{c_{3}^{2} - c_{1}^{2}} \\{c_{4}^{2} - c_{1}^{2}}\end{pmatrix}}}},} & \;\end{matrix}$

and where the Cartesian coordinate of {right arrow over (r)}_(i) is(r_(i)(x), r_(i)(y), r_(i)(z)). the inverse of R is R⁻¹. We form thequantity {right arrow over (P)}^(T) {right arrow over (P)} from Eq. (31)to obtain,

$\begin{matrix}{{{{{\overset{\rightarrow}{P}}^{T}\overset{\rightarrow}{P}} \equiv {\overset{\rightarrow}{P}}^{2}} = {\frac{\alpha_{1}}{4} - {\alpha_{2}t_{0}} + {\left( {\alpha_{3} - \alpha_{4}} \right)t_{0}^{2}} + {2\alpha_{5}t_{0}^{3}} + {\alpha_{6}t_{0}^{4}}}},} & (33)\end{matrix}$

where

α₁≡(R ⁻¹ {right arrow over (b)} _(e))^(T)(R ⁻¹ {right arrow over (b)}_(e))

α₂≡(R ⁻¹ {right arrow over (b)} _(e))^(T)(R ⁻¹{right arrow over(ƒ)}_(e))

α₃≡(R ⁻¹{right arrow over (ƒ)}_(e))^(T)(R ⁻¹{right arrow over (ƒ)}_(e))

α₄≡(R ⁻¹ {right arrow over (b)} _(e))^(T)(R ⁻¹ {right arrow over (g)}_(e))

α₅≡(R ⁻¹ {right arrow over (g)} _(e))^(T)(R ⁻¹ {right arrow over (g)}_(e))   (34)

Expanding Eq. (28), we get,

∥{right arrow over (P)}∥ ² =c ₁ ²(T ₁ ²−2t ₀ T ₁ +t ₀ ²),   (35)

which is substituted into Eq. (33) to yield a quartic equation in t₀,

$\begin{matrix}{{{\alpha_{6}t_{0}^{4}} + {2\alpha_{5}t_{0}^{3}} + {\left( {\alpha_{3} - \alpha_{4} - c_{1}^{2}} \right)t_{0}^{2}} + {\left( {{2T_{1}c_{1}^{2}} - \alpha_{2}} \right)t_{0}} + \frac{\alpha_{1}}{4} - {c_{1}^{2}T_{1}^{2}}} = 0.} & (36)\end{matrix}$

This can be solved for t₀ analytically or with a root finder. Validroots for t₀ are substituted into Eq. (31) to obtain the location of theprimary point, {right arrow over (P)}. Some of the roots for Eq. (36)can be invalid because they do not satisfy Eq. (24). This occurs becausethe algebra involves squaring quantities, and one must remain on thecorrect branch when trying to un-do the squaring. One can guarantee thatthe solution for t₀ is on the correct branch by requiring each real andnon-negative solution of Eq. (36) to obey Eq. (24). Choosing the correctbranch for the solution comes up again when discussing the geometry ofisosigmachrons (below).

The solution to Eqs. (31) and (36) can lead to ambiguous locations evenafter removing solutions not obeying Eq. (24). In this case, onerequires another datum to decide which ambiguous primary point solutionis correct. One can use the location of the source with each ambiguousprimary point location to predict T_(i), i=1, 2, 3, 4. Let thepredictions be denoted {tilde over (T)}_(i). One can then choose theambiguous primary point location that minimizes,

$\begin{matrix}{F \equiv {\sum\limits_{i = 1}^{4}{\left( {T_{i} - {\overset{\sim}{T}}_{i}} \right)^{2}.}}} & (37)\end{matrix}$

There are many other objective functions that can be used beside F.Alternatively, one could also use the travel time measurement from afifth receiver to find which ambiguous primary point yields a predictedtime closest to that at a fifth receiver.

One can summarize the ambiguities as follows. For three spatialdimensions, one starts with four receivers to determine the location ofa primary point in the absence of errors. If ambiguous solutions occur,one can use either the location of the source or the data from a fifthreceiver to decide which ambiguous solution for the primary point iscorrect.

When these equations are utilized for ellipsoidal location, i.e. allspeeds of propagation are identical, ambiguous solutions can occur. Asabove, one can use either the location of the source or the data from afifth receiver to decide which ambiguous solution for the primary pointis correct.

In two spatial dimensions, one starts with three receivers to determinethe location of a primary point in the absence of errors. If ambiguoussolutions occur, one can use either the location of the source or thedata from a fourth receiver to decide which ambiguous solution for theprimary point is correct.

For two-dimensional locations, one drops station {right arrow over (r)}₄and the datum T₄ from the equations above to obtain a solution for theprimary point.

-   -   -   2. Solution Using Three-Receiver Constellation and Source            Location

There is another way to find an analytical solution for the location ofthe primary point that uses three receivers instead of four in threedimensions. One uses an equation not used above, namely,

∥{right arrow over (P)}−{right arrow over (s)}∥ ² =c ₀ ² t ₀ ²,   (38)

and subtracting Eq. (28) from Eq. (38) yields,

∥{right arrow over (P)}−{right arrow over (s)}∥ ² −∥{right arrow over(P)}∥ ² =c ₀ ² t ₀ ² −c ₁ ²(T ₁ −t ₀)².

This simplifies to,

$\begin{matrix}{{{\overset{\rightarrow}{s}}^{T}\overset{\rightarrow}{P}} = {{\frac{1}{2}{\overset{\rightarrow}{s}}^{2}} + \frac{c_{1}^{2}T_{1}^{2}}{2} - {c_{1}^{2}T_{1}t_{0}} + {\frac{1}{2}\left( {c_{1}^{2} - c_{0}^{2}} \right){t_{0}^{2}.}}}} & (39)\end{matrix}$

Without loss of generality, use Eq. i=2, 3 from Eq. (29) in addition toEq. (39) to get,

Q{right arrow over (P)}=½{right arrow over (β)}−t ₀{right arrow over(ρ)}−t ₀ ²{right arrow over (γ)},   (40)

where,

$\begin{matrix}{{Q \equiv \begin{pmatrix}{r_{2}(x)} & {r_{2}(y)} & {r_{2}(z)} \\{r_{3}(x)} & {r_{3}(y)} & {r_{3}(z)} \\{s(x)} & {s(y)} & {s(z)}\end{pmatrix}};} & (41)\end{matrix}$

and,

$\begin{matrix}{{{\overset{\rightarrow}{\beta} \equiv \begin{pmatrix}{{{\overset{\rightarrow}{r}}_{2}}^{2} + \left( {{c_{1}^{2}T_{1}^{2}} - {c_{2}^{2}T_{2}^{2}}} \right)} \\{{{\overset{\rightarrow}{r}}_{3}}^{2} + \left( {{c_{1}^{2}T_{1}^{2}} - {c_{3}^{2}T_{3}^{2}}} \right)} \\{{\overset{\rightarrow}{s}}^{2} + {c_{1}^{2}T_{1}^{2}}}\end{pmatrix}};}{{\overset{\rightarrow}{\rho} \equiv \begin{pmatrix}{{T_{1}c_{1}^{2}} - {T_{2}c_{2}^{2}}} \\{{T_{1}c_{1}^{2}} - {T_{3}c_{3}^{2}}} \\{T_{1}c_{1}^{2}}\end{pmatrix}};}{{\overset{\rightarrow}{\gamma} \equiv {\frac{1}{2}\begin{pmatrix}{c_{2}^{2} - c_{1}^{2}} \\{c_{3}^{2} - c_{1}^{2}} \\{c_{0}^{2} - c_{1}^{2}}\end{pmatrix}}},}} & (42)\end{matrix}$

and where the Cartesian coordinates of the source are {right arrow over(s)}≡(s(x), s(y), s(z)). The solution for the primary point, {rightarrow over (P)}, is found in a manner analogous to that in the previoussection. Eq. (40) is multiplied by Q⁻¹, which usually exists,

{right arrow over (P)}=½Q ⁻¹ {right arrow over (β)}−t ₀ Q ⁻¹ {rightarrow over (ρ)}−t ₀ ² Q ⁻¹{right arrow over (γ)},   (43)

and then one forms {right arrow over (P)}^(T) {right arrow over(P)}≡∥{right arrow over (P)}∥² and equates this to Eq. (35) to yield aquartic equation in t₀,

$\begin{matrix}{{{A_{6}t_{0}^{4}} + {2A_{5}t_{0}^{3}} + {\left( {A_{3} - A_{4} - c_{1}^{2}} \right)t_{0}^{2}} + {\left( {{2T_{1}c_{1}^{2}} - A_{2}} \right)t_{0}} + \frac{A_{1}}{4} - {c_{1}^{2}T_{1}^{2}}} = 0.} & (44)\end{matrix}$

where,

A ₁≡(Q ⁻¹{right arrow over (β)})^(T)(Q ⁻¹{right arrow over (β)})

A ₂≡(Q ⁻¹{right arrow over (β)})^(T)(Q ⁻¹{right arrow over (ρ)})

A ₃≡(Q ⁻¹{right arrow over (ρ)})^(T)(Q ⁻¹{right arrow over (ρ)})

A ₄≡(Q ⁻¹{right arrow over (β)})^(T)(Q ⁻¹{right arrow over (γ)})

A ₅≡(Q ⁻¹{right arrow over (ρ)})^(T)(Q ⁻¹{right arrow over (γ)})

A ₆≡(Q ⁻¹{right arrow over (γ)})^(T)(Q ⁻¹{right arrow over (γ)})   (45)

The valid solution for t₀ is substituted into Eq. (43) to yield thelocation of the primary point. Some ambiguous solutions can beeliminated by ensuring that it gives a location on the correct branch bymaking it obey Eq. (24). If this does not resolve the ambiguity, oneneeds the travel time collected at a fourth receiver to resolve theambiguity. The ambiguity is resolved by seeing which ambiguous locationyields a travel time closest to that measured by a fourth receiver.

When these equations are utilized for ellipsoidal location, i.e. allspeeds of propagation are identical, ambiguous solutions can occur andcan be resolved in the same way as above.

For two-dimensional isosigmachronic locations, one drops station {rightarrow over (r)}₃ and the datum T₃ from the equations above to obtain asolution for the primary point.

-   -   B. Geometry of Isosigmachrons

An isosigmachron is a geometrical shape that facilitates an analyticalsolution for location. Its geometry is described here.

Let l₀ and l₁ denote distances from Cartesian coordinates (−a, 0, 0) and(a, 0, 0). An isosigmachron is defined to be the locus of points, (x, y,z) satisfying,

$\begin{matrix}{{{{\frac{l_{0}}{c_{0}} + \frac{l_{1}}{c_{1}}} = T};}{c_{0} \neq c_{1}}} & (46)\end{matrix}$

where the speeds of propagation are c₀ and c₁ from (−a, 0, 0) to (x, y,z) and from (x, y, z) to (a, 0, 0) respectively, and where l₀=√{squareroot over ((x+a)²+y²+z²)} and l₁=√{square root over ((x−a)²+y²+z²)}.When the speeds of propagation are equal, one no longer has anisosigmachron, but rather an ellipsoid. The geometry of isosigmachronswill be described in the x−y plane without loss of generality. Rotatinga two-dimensional isosigmachron about the line passing through (−a, 0)and (a, 0) yields the three-dimensional isosigmachron.

With T given by Eq. (24), the Cartesian coordinates of thetwo-dimensional isosigmachron obey,

$\begin{matrix}{{\frac{\sqrt{\left( {x + a} \right)^{2} + y^{2}}}{c_{0}} + \frac{\sqrt{\left( {x - a} \right)^{2} + y^{2}}}{c_{1}}} = {T.}} & (47)\end{matrix}$

The simplest isosigmachron is one in which c₀ and c₁ are independent ofcoordinate. Making this assumption, one employs algebra to yield apolynomial that can be written in two equivalent ways. The first way is,

d ₄ x ⁴ +d ₃ x ³ +d ₂ x ² +d ₁ x+d ₀=0,   (48)

where,

$\begin{matrix}{\mspace{79mu} {{d_{4} = {{- \left( {c_{0} - c_{1}} \right)^{2}}\left( {c_{0} + c_{1}} \right)^{2}}}\mspace{79mu} {d_{3} = {4{a\left( {c_{0} - c_{1}} \right)}\left( {c_{0} + c_{1}} \right){\left( {c_{0}^{2} + c_{1}^{2}} \right)/c_{1}^{4}}}}{d_{2} = {\frac{- 2}{c_{1}^{4}}\left( {{3a^{2}c_{1}^{4}} + {y^{2}c_{1}^{4}} + {3c_{0}^{4}a^{2}} + {c_{0}^{4}y^{2}} - {c_{0}^{2}T^{2}c_{1}^{4}} - {c_{0}^{4}T^{2}c_{1}^{2}} + {2c_{0}^{2}c_{1}^{2}a^{2}} - {2c_{0}^{2}c_{1}^{2}y^{2}}} \right)}}{d_{1} = {\frac{4a}{c_{1}^{4}}\left( {c_{0} - c_{1}} \right)\left( {c_{0} + c_{1}} \right)\left( {{c_{0}^{2}a^{2}} + {c_{0}^{2}y^{2}} - {c_{0}^{2}T^{2}c_{1}^{2}} + {a^{2}c_{1}^{2}} + {y^{2}c_{1}^{2}}} \right)}}{d_{0} = {\frac{- 1}{c_{1}^{4}}\left( {{c_{0}^{2}T^{2}c_{1}^{2}} + {a^{2}c_{1}^{2}} + {y^{2}c_{1}^{2}} + {2c_{0}a^{2}c_{1}} + {2c_{1}y^{2}c_{0}} + {c_{0}^{2}a^{2}} + {c_{0}^{2}y^{2}}} \right)\left( {{{- c_{0}^{2}}T^{2}c_{1}^{2}} + {a^{2}c_{1}^{2}} + {y^{2}c_{1}^{2}} - {2c_{0}a^{2}c_{1}} - {2c_{1}y^{2}c_{0}} + {c_{0}^{2}a^{2}} + {c_{0}^{2}y^{2}}} \right)}}}} & (49)\end{matrix}$

and the second is,

e ₄ y ⁴ +e ₂ y ² +e ₀=0   (50)

where,

     e₄ = −(c₀ − c₁)²(c₀ + c₁)²/c₁⁴$e_{2} = {\frac{- 2}{c_{1}^{4}}\left( {{x^{2}c_{1}^{4}} + {a^{2}c_{1}^{4}} + {c_{0}^{4}x^{2}} - {2c_{0}^{4}{xa}} + {c_{0}^{4}a^{2}} + {2{xac}_{1}^{4}} - {c_{0}^{2}T^{2}c_{1}^{4}} - {c_{0}^{4}T^{2}c_{1}^{2}} - {2x^{2}c_{0}^{2}c_{1}^{2}} - {2c_{0}^{2}a^{2}c_{1}^{2}}} \right)}$$e_{0} = {\frac{- 1}{c_{1}^{4}}\left( {{{- c_{0}}c_{1}T} + {c_{0}x} + {c_{1}x} - {c_{0}a} + {c_{1}a}} \right)\left( {{c_{0}c_{1}T} + {c_{0}x} - {c_{1}x} - {c_{0}a} - {c_{1}a}} \right)\left( {{{- c_{0}}c_{1}T} + {c_{0}x} - {c_{1}x} - {c_{0}a} - {c_{1}a}} \right)\left( {{c_{0}c_{1}T} + {c_{0}x} + {c_{1}x} - {c_{0}a} + {c_{1}a}} \right)}$

The polynomial in x can be simplified by substituting,

w≡y²,   (51)

in Eq. (50) to yield,

e ₄ w ² +e ₂ w+e ₀=0.   (52)

When e₄ is not zero,

$\begin{matrix}{w = {\frac{{- e_{2}} \pm \sqrt{e_{2}^{2} - {4e_{0}e_{4}}}}{2e_{4}}.}} & (53)\end{matrix}$

After choosing values of x for which w is real and non-negative, one canobtain the solution for y as,

y=±√{square root over (w)},   (54)

where one needs to choose the proper sign ± in Eq. (53) such that Eq.(24) is obeyed.

When e₄ is zero and when e₂ is not zero, the solution is,

y=±√{square root over (−e₀ /e ₂)}; e ₄=0 and e ₂≢0 and −e ₀ /e ₂≧0.  (55)

The x intercepts for the isosigmachron are given by,

$\begin{matrix}{x = \left\{ \begin{matrix}\frac{{c_{0}c_{1}T} + {a\left( {c_{0} - c_{1}} \right)}}{c_{0} + c_{1}} & {{{for}\mspace{14mu} x} \geq a} \\\frac{{{- c_{0}}c_{1}T} - {a\left( {c_{1} - c_{0}} \right)}}{c_{0} + c_{1}} & {{{for}\mspace{14mu} x} \leq {- a}}\end{matrix} \right.} & (56)\end{matrix}$

which reduce to the ellipse's x intercepts of ±cT/2 when c₀=c₁=c. Theisosigmachron looks like a skewed ellipse (FIG. 15).

-   -   C. Ellipsoidal and Isosigmachronic Location Via Sequential Bound        Estimation

Suppose the signal initiating at {right arrow over (s)} is received atfewer stations than needed to obtain an unambiguous solution for thelocation of the primary point {right arrow over (P)}. Next, supposeanother signal is transmitted from another location, {right arrow over(s)}, and received, after encountering {right arrow over (P)}, onpossibly a different set of stations. Consider a situation where theprimary point is moving and data of the type in Eq. (24) are measuredsequentially in time.

One can use Sequential Bound Estimation to estimate the location of themobile primary point in the following way. Without loss of generality,assume the geometry is two-dimensional in the x−y Cartesian plane.

-   -   1. The datum of travel time from the first receiver, T₁(t), is        used to constrain the primary point within a cloud of locations        in the plane by using an analytical solution for an ellipse or        an isosigmachron. Here, T_(i) is given by Eq. (24) and t is        geophysical time (the time the measurement was taken). The cloud        of points is found by varying all relevant variables with a        priori probability distributions. Such variables are noise in        the measurement, the effective speed of propagation including        currents if any, and the locations of the receivers.    -   2. Find spatial bounds for the location of the primary point        from the cloud of locations.    -   3. A model is used to expand the spatial bounds for the primary        point between times t and t+δt, the time at which the next        datum, T₂, is measured. The expansion of the bounds is made in a        manner consistent with the velocity of the primary point.    -   4. The datum from T₂(t+δt) is used to constrain the location of        the primary point with a new cloud of locations by using an        analytical solution for an ellipse or an isosigmachron like in        item 1. The probability distributions of the variables are        started anew with a priori distributions but using possibly new        bounds as obtained from item 3.

It can be seen that this procedure is sequential, and can be continueduntil all travel time data are utilized. The procedure is SequentialBound Estimation because each datum is assimilated using a prioridistributions modified by bounds established from previous steps andmodels.

It can also be seen that one can additionally utilize all bounds fromthe previous step to constrain the probability distributions of thevariables for the next step. For example, suppose the a priori limitsfor the location of a receiver are larger than the cloud of receiverlocations that fit the data. Then one has estimated more accurate limitsfor that receiver's location. The same procedure can be used to improveestimates for any of the variables, including error in the data,receiver locations, speeds of signal propagation, and locations of thesource and primary point. At the end of each step, one has a probabilitydistribution for each of these variables from which any confidence limitor other function of the probability distribution can be obtained. Ifany of these variables have uncertainties that change with time, or ifthere is a model that can be used to estimate these variables, thoseconstraints are simple to impose on the variables to transition them tothe time of the next datum.

One can use measurements of the direction of the incoming signal at areceiver to constrain the locations of the primary point in thefollowing way. Suppose that a receiver is equipped with an array thatmeasures the incoming signal between the bounds of θ₁ and θ₂ degreeswith a probability distribution F(θ). Suppose that the locus of pointsalong which the primary point resides obeys the equation y(x, c₀, c₁,{right arrow over (r)}₁, {right arrow over (s)}) where {right arrow over(c)}₀ is the effective speed of propagation from {right arrow over (s)}to (x, y(x)) (a location on an ellipse or isosigmachron), and {rightarrow over (c)}₁ is the effective speed between (x, y(x)) and {rightarrow over (r)}₁. The function, y(x, c₀, c₁, {right arrow over (r)}₁,{right arrow over (s)}), describes an ellipse or an isosigmachron forany particular set of simulated random variables, {x, c₀, c₁, {rightarrow over (r)}₁, {right arrow over (s)}}. The prior directionalinformation from F(θ) can be translated into a prior distribution forthe variable, x, using,

F(θ_(j)≦θ≦θ_(k))=F(x(θ_(j) ,c ₀ ,c ₁ ,{right arrow over (r)} ₁ ,{rightarrow over (s)})≦x≦x(θ_(k) ,c ₀ ,c ₁ ,{right arrow over (r)} ₁ ,{rightarrow over (s)})),   (57)

where x(θ, c₀, c₁, {right arrow over (r)}₁, {right arrow over (s)}) isthe value of x given the set of random variables {θ, c₀, c₁, {rightarrow over (r)}₁, {right arrow over (s)}}. It is assumed that θ_(j) andθ_(k) represent a small enough interval within θ₁ to θ₂ so thatsufficiently accurate estimates can be obtained for the right side inEq. (57). The function, x(θ, c₀, c₁, {right arrow over (r)}₁, {rightarrow over (s)}), can be obtained numerically or analytically. Forexample, when y(x, c₀, c₁, {right arrow over (r)}₁, {right arrow over(s)}) is described by an ellipse,

$\begin{matrix}{{x = {{{{\overset{\rightarrow}{s} - {\overset{\rightarrow}{r}}_{1}}}/2} + \frac{{a\left( {1 - ^{2}} \right)}\cos \; \theta}{1 + {e\; \cos \; \theta}}}},} & (58)\end{matrix}$

where the semi-major and minor axes of the ellipse are a and brespectively, and the eccentricity is e≡√{square root over (1−b²/a²)}.The x axis in Eq. (58) is defined to pass through {right arrow over (s)}and {right arrow over (r)}₁. If this does not correspond to the axes ofthe original problem, it can be transformed back to the unshifted andunrotated Coordinate frame using a transformation of coordinates. Tofurther help the reader understand how to use Eq. (58), suppose it isknown, for simplicity, that F(θ) is distributed according to a truncatedGaussian distribution. One can generate a truncated Gaussiandistribution between θ₁ and θ₂ with some mean, and put these randomvariables for θ into Eq. (58) to obtain the corresponding randomvariables for x. Thus the priori distribution of arrival angle isincorporated into the distribution of random variables in x, which inturn is used to generate a consistent set of points for the cloud oflocations for the primary point.

Applications of this idea can be used by the Navy who sometime locateships in the sea by measuring the travel time of sound from a source toa point of interest such as a submarine and back to an acousticreceiver. There are other applications for location of this type thatoccur in medicine when one uses sound to insonify the body, and thetravel time to a receiver is measured after reflecting from the body'sinterior. Another application includes the use of sound for geophysicalmapping of the Earth, or for the exploration of oil.

It is important to note that no linear approximation has been made andone is sequentially processing data without any difficulty for realisticprobability distributions for all the variables that come into play.

IX. Ellipsoidal and Isosigmachronic Location: Experiment

On 12 Apr. 2003 an experiment was conducted at 6 Derring Dale Rd,Radnor, Pa. to collect data for the purpose of verifying the new methodsfor locating a primary point using ellipsoidal or isosigmachronictechniques. An approximate Cartesian x′−y′ grid was drawn using chalk ona blacktop with the x′ and y′ axes extending for 9.1 and 6.7 mrespectively. The angle between the x′ and y′ axes was estimated asfollows. The distance between a point on the x′ axis to a point on they′ axis was measured with a steel tape, and then the law of cosines wasused derive the angle by also using the distances of the sides of thetriangle along the x′ and y′ axes respectively. The procedure was donefor two sets of measurements to yield estimates of 89.48° and 89.22° forthe angle between the primed axes. Their average of 89.35° degrees wasused for the remaining calculations. Conversion to a Cartesian x−y gridwas made from the primed reference frame using trigonometry.

The locations of the source, primary point, and each of five stationswere marked with chalk on the grid. A steel tape measure was used tomeasure distances along the primed axes and to measure the locations ofthe source, primary point, and stations. The locations of points not onthe x′ or y′ axes were estimated by measuring their distances from threedifferent points on the x′ or y′ axes, and then using the law of cosinesto provide two different estimates of the same location. The differencebetween the two estimates provided information on the accuracy of eachlocation. Remaining values for locations are converted to the Cartesiangrid using trigonometry (Table VIII).

TABLE VIII LOCATION (x, y) (m) SOURCE (0.0034, 0.305) PRIMARY POINT(4.538, 4.971) STATION 1 (1.219, 0) STATION 2 (5.182, 0) STATION 3(9.144, 0) STATION 4 (0.0692, 6.096) STATION 5 (7.373, 4.185)

Errors for the locations in Table VIII are modeled as being mutuallyuncorrelated Gaussian random variables with means of zero and standarddeviations of 0.06 m, 0.06 m, and 0.15 m respectively for the x, y, andz Cartesian coordinates respectively except as follows. The location ofstation one has zero error as it defines the origin of the coordinatesystem. All errors are truncated at one standard deviation.

The time for signal propagation between the source and primary pointplus primary point to each station was measured by timing the author'swalking time-of-travel along each route with a Radio Shack model 63-5017stopwatch (Table IX). The time between the source and primary point wasmeasured five times. The walking time was measured between the primarypoint and each station. Each of these measurements was taken four orfive times. Repetitions of the times were important for estimatingerrors. To eliminate the changes in transit times that arise fromwalking the corner at the primary point, the inventor measured the timesof each walk in a straight line and added two timings together to getthe time from the source to primary point plus primary point to eachstation. To further reduce errors due to the change in speed from astandstill, the inventor started the walk a few steps before eachstarting point and walked through each ending point always striving tomaintain a steady speed. The inventor did not look at the stopwatch tostart and end its interval timer. Rather, the inventor looked at thechalk marks on the ground to initiate start and stop times. The valuesin Table IX are averages of the four to five separate measurements. Thelargest deviation in walking time between the four to five separatemeasurements was 0.2 s. The errors in the travel times for the locationalgorithm were modeled as mutually uncorrelated Gaussian randomvariables with means given by Table IX and standard deviations of 0.22s. The distributions were truncated at one standard deviation.

To convert times to distances, it is necessary to estimate the speed ofwalking. The author measured his speed over a straight line of 7.925 mten separate times, again using a stopwatch. The average speed was 1.481m/s with largest deviation being 0.098 m/s. The location algorithmassumed the speed of walking was 1.481 m/s with a Gaussian random errorof 0.15 m/s and truncated at one standard deviation. Thus the error inspeed was probably slightly overestimated for the sake of caution.

TABLE IX DESTINATION TIME (s) STATION 1 8.375 STATION 2 7.695 STATION 38.814 STATION 4 7.465 STATION 5 6.355

-   -   A. Four Receiver Constellation

There are five ways of choosing four stations from a total of the fivein Table VIII. For each four-receiver (station) constellation, up to500,000 Monte-Carlo simulations were run to estimate locations for theprimary point (Table VIII) from the measured times of travel (Table IX)using Eqs. (31) and (36). The a priori bounds for the primary point wereset at x: 0 to 20 m; y: 0 to 30 m; z: −0.2 to +0.2 m. Forisosigmachronic location, the 100% confidence limits for the primarypoint are x: 3.81 to 5.29 m; y: 3.77 to 7.17 m; z: −0.20 to 0.29 m. The95% confidence limits for the primary point are x: 4.11 to 5.10 m; y:4.06 to 5.95 m; z: −0.19 to 0.19 m. Both are consistent with thelocation of the primary point (Table VIII).

For ellipsoidal location, the 100% confidence limits for the primarypoint are x: 4.14 to 5.09 m; y: 4.24 to 5.98 m; z: −0.2 to +0.2 m. The95% confidence limits for the primary point are x: 4.27 to 4.93 m; y:4.35 to 5.61 m; z: −0.19 to +0.19 m. Both are consistent with thelocation of the primary point (Table VIII). In general, however,ellipsoidal location need not be consistent with the measured value ofthe primary point when the speed of propagation is inhomogeneous.

-   -   B. Three-Receiver Constellation and Source Location

There are ten ways of choosing three stations from a total of five inTable VIII. For each three-receiver (station) constellation plus sourcelocation, up to 500,000 Monte-Carlo simulations were run to estimatelocations for the primary point (Table VIII) from the measured times oftravel (Table IX) using Eqs. (43) and (44). The a priori bounds for theprimary point were set at x: 0 to 20 m; y: 0 to 30 m; z: −0.2 to +0.2 m.For isosigmachronic location, the 100% confidence limits for the primarypoint are x: 3.78 to 5.22 m; y: 3.99 to 5.74 m; z: −0.20 to +0.20 m. The95% confidence limits for the primary point are x: 4.01 to 5.00 m; y:4.22 to 5.56 m; z: −0.19 to +0.19 m. Both are consistent with thelocation of the primary point (Table VIII).

For ellipsoidal location, the 100% confidence limits for the primarypoint are x: 4.16 to 5.08 m; y: 4.25 to 5.73 m; z: −0.20 to +0.20 m. The95% confidence limits for the primary point are x: 4.22 to 4.98 m; y:4.36 to 5.60 m; z: −0.19 to +0.19 m. Both are consistent with thelocation of the primary point (Table VIII). In general, however,ellipsoidal location need not be consistent with the measured value ofthe primary point when the speed of propagation is inhomogeneous.

-   -   C. Sequential Location from Under-Determined Systems

The method given in Section VIII, subsection C above is applied to thisexperiment to demonstrate its efficacy. The same errors and locationsare assumed as above.

First, the travel time at R1 is combined with the estimate of thedirection toward the primary point to obtain the cloud of locationsgiven by small dots in FIG. 16. A two-dimensional approximation is usedfor this experiment so the cloud of points come from a two-dimensionalisosigmachron. The direction from R1 toward the primary point wasestimated by eye to be within 15° to 40° degrees True where the y axisis assumed to point North. The a priori distribution of this angle isassumed for simplicity to be uniformly distributed between these angularlimits. Eq. (58) is used to generate random values of x in a rotatedcoordinate frame from the uniformly distributed values of θ. Comparedwith a priori limits for the primary point of x: 0 to 20 m, y: 0 to 30m, the cloud in FIG. 16 have limits of x: 4.05 to 5.50 m and y: 3.90 to5.90 m.

Next, Sequential Bound Estimation is used to estimate the next cloud ofprimary point locations. The a priori Cartesian limits are those fromthe end of the last step, namely x: 4.05 to 5.50 m and y: 3.90 to 5.90m. Then, the travel time and directional information from R2 is used tofind realizations of primary point locations (FIG. 17). The directiontoward the primary point is estimated by eye from R2 to be between 340°and 358° degrees True. The cloud in FIG. 17 has Cartesian limits of x:4.11 to 5.09 m, and y: 3.91 to 5.82 m. These are less than the limitsbefore and are consistent with the location of the primary point (TableVIII).

X. Ellipsoidal and Isosigmachronic Location: Concise Description

This section concisely describes the inventions as they relate toellipsoidal and isosigmachronic location and Sequential BoundEstimation. The word “station” can refer to either a source or areceiver as discussed in the previous discussions on ellipsoidal andisosigmachronic location.

Referring to FIG. 4 for isosigmachronic location, travel times arecollected from a constellation of stations 200. Estimates for the speedof the signal are obtained between the primary point and each station210. When advection of the medium does not significantly affect thedelay of the signal between the stations and the primary point, onecomputes the location of the primary point using an analytical formula230. When isosigmachronic location is implemented the effective speed ofpropagation is different among some of the paths. It is possible thatthe analytical equation for isosigmachronic location does not yield areal valued solution. This can happen when estimates of the speeds ofpropagation and station locations are not close enough to their truevalues.

Attention now turns to FIG. 5 for a discussion of iteration. If theeffects of advection due such things as winds or currents significantlyaffect the time of travel between the primary point and a station, abound for the location of the primary point is established 240. Becausethe actual location of the primary point is unknown, the precise traveltime change between the primary point and each station due to advectionis initially unknown. So a guess is made within the bounds for thelocation of the primary point 250. The effective speed between theprimary point and each station is computed using the vector projectionof the advection along the straight line between the primary point andeach station. The effective speed also includes the non-advectivecomponent of speed 260. For example, the speed of sound may be 330 m/s,while the effects of the winds are +1 m/s from the primary point tostation one. The effective speed from the primary point to station oneis therefore 330+1=331 m/s.

The location of the primary point is estimated using an analyticalformula given the effective speeds. If this location is close enough tothe most recent guess for this location, one exits the iterativeprocedure with a solution for the location of the primary point 280.Otherwise, the analytical solution for the location of the primary pointis used to compute another set of effective speeds to each station untilthe convergence criterion is either met 280 or exceeded 290. Inpractical application of this idea, convergence is usually obtainedafter one or two iterations.

Like Sequential Bound Estimation, this algorithm can detect problemswith the data, or assumptions about the variables by yielding nosolutions for the location of the primary point. The process stops ifsuch problems are encountered. Problems can be found at step 230.

In a related invention now refer to FIG. 18. In this case, the goal isto estimate any function of the probability distribution of any variableplaying a role in isosigmachronic location. The variables are thearguments of a nonlinear function for location and the value of anonlinear function. Travel times are estimated from a number of stationsequal to or exceeding the number in a constellation 400. Prior boundsare established for all variables 410. Optionally, a database of traveltimes is used to establish possibly smaller bounds for the location ofthe primary point 460, 470.

If the number of stations equals the number in a constellation,probability distributions are assigned to each variable within priorbounds except for the location of the primary point that is nonlinearlyrelated to the other variables 420. Realizations of primary points areobtained by applying an analytical solution for the location of theprimary point to realizations of arguments of the nonlinear functionusing a Monte-Carlo technique 430. Realizations leading to primarypoints outside of assigned bounds are excluded from furtherconsideration 430. The remaining realizations are used to form anyfunction of the probability distribution of any variable 450.

Realizations of primary points here includes the possibility thatadvection can alter the travel times enough to represent a significanteffect. In this case, one iterates the analytical solution untilconvergence is achieved or until convergence is not achieved, and therealization is discarded. Details of iterating the analytical solutionare described in FIG. 5.

Like Sequential Bound Estimation, this algorithm can detect problemswith the data, or problems concerning assumptions about the variables byyielding no solutions for the location of the primary point. The processstops if no realizations are obtained for the primary point 460, 480,430. The algorithm halts execution if posterior bounds are inconsistentwith prior bounds 480 from different steps of Sequential BoundEstimation.

Returning again to FIG. 18, another related invention is estimating anyfunction of the probability distribution of any variable playing a rolein ellipsoidal location. The method is the same as used forisosigmachronic location except as follows.

The effective speed of propagation between the primary point and eachstation must be the same value for any particular realization.Naturally, the single value of the effective speed used for realizationj can be allowed to differ from that for realization k at the stagewhere the Monte-Carlo technique is implemented. If travel times aresignificantly affected by advection, the bounds for the effective speedcan be modified to attempt to account for some of this effect asfollows. Let the magnitude of velocity due to advection be |U_(max)|.When bounds are chosen for the effective speed for each path, instead ofusing upper and lower bounds respectively for the unadvected speeds ofpropagation, ć and {hacek over (c)}, one can use bounds for theeffective speed, {circumflex over (μ)} and {hacek over (μ)}, such as,

{circumflex over (μ)}=ĉ+|U _(max)|  (59)

{hacek over (μ)}={hacek over (c)}−|U _(max)|  (60)

Naturally, one uses equations such as these only for signals that areaffected by advection like sonic signals, since the speed ofelectromagnetic waves is unaffected by advection. It is also importantto note that the analytical solutions for ellipsoidal locations existthat do not require iteration of the analytical solution. Two suchsolutions are given Eqs. (31,36) and (43,44) where all coefficients areevaluated with the same value for the effective speed of propagation.Other methods for finding the primary point by intersecting ellipses arefound elsewhere (Eberly, Magic Software, Inc.,http://www.magic-software.com, Oct. 10, 2000). In this invention, thephrase “primary function” is the function whose arguments include thesums of travel times, the locations of stations, and the speed ofpropagation. This function may be linear (Eberly, 2000) or nonlinear.This function may also include other variables.

In a related invention now refer to FIG. 19. The travel time of a signaland its incoming direction from a primary point are estimated at astation 600. Prior bounds are established for all variables includingthe speed of propagation, advective effects such as currents (ifsignificant), the location of the station, the location of the source ofthe signal, and errors of the data 610. For ellipsoidal location, oneaccounts for significant effects of advection on the travel time usingthe method in the previous paragraph. For isosigmachronic location, onemay use the iterative procedure from FIG. 5 to iterate analyticalsolutions for location. These are the variables that are “functionallyaffecting” the location of the primary point.

Probability distributions are assigned to each variable except for thelocation of the primary point which is related to the other variablesthrough a primary function 620. Statistical information regarding thedirection of the incoming signal can be transformed to a Cartesiancoordinate for later use via expressions such as Eq. (57) or (58). Othercoordinate systems could be used to make this transformation.Realizations for the location of the primary point are simulated fromthe probability distributions of the other variables using a Monte-Carlotechnique 630. Ellipsoidal locations use identical speeds of signalpropagation for all paths within any one realization, though that speedmay differ from one realization to the next. For isosigmachroniclocations, the speed of signal propagation is allowed to differ amongdifferent paths within any single realization.

The only realizations that are used are those falling within the spatialbounds for the primary point 630. The posterior bounds for the primarypoint are estimated from the realizations 650. Optionally, one canobtain any function of the probability distributions of any variable at660. If there are more data to assimilate, one employs Sequential BoundEstimation 670. Otherwise, the algorithm exits with the requestedinformation 680. The procedure can be used with either ellipsoidal orisosigmachronic geometry.

This algorithm can detect problems with the data, or assumptions aboutthe variables by yielding no solutions for the location of the primarypoint. These problems stop the algorithm 630 and 670. The algorithmstops if bounds associated with one step are inconsistent with thosefrom another step 670.

The procedure in FIG. 19 has application to locating ships and otherobjects at sea with active sonar. When implemented for a two-dimensionalgeometry, the algorithm requires about 10 seconds to run on a personalcomputer. The three-dimensional implementation of ellipsoidal orisosigmachronic location requires significantly more computer time.

XI. Description of Operation and General Content of Computer ProgramListings

The following software is submitted as part of this application as acomputer program listing appendix on CDROM with a duplicate copy.

-   -   1. drawisosigma.m: Matlab code which draws ellipse and        isosigmachron in two dimensions. Parameters for the geometrical        shapes are defined at the top of the program. To run the        program, start matlab and type the name of this script file.    -   2. isodiachron.m: Matlab code which draws an isodiachron and a        hyperbola in two dimensions. Parameters for the geometrical        shapes are defined near the top of the program. To run the        program, start matlab and type the name of this script file.    -   3. boundP_(—)2d.m: Matlab main script file. Does 2-D ellipsoidal        or isosigmachronic location for one receiver at a time. The        receiver has directional information about the primary point.        Program has several ascii input files that supply data and modes        for its operation. This program is used to do sequential bound        estimation if desired. Program starts by starting matlab and        typing boundP_(—)2d.m. It calls the following matlab script        files:        -   (a) comb1.m: Computes the number of ways of selecting M            things from K bins without replacement.        -   (b) genarr.m: Generates arrangements of K things taken M at            a time.        -   (c) getisosigma.m: This generates (x,y) points on the            isosigmachron or ellipse.        -   (d) getx_rv.m: Gets random values of x from directional            information to the primary point from a receiver. These            values of x are later used to obtain (x,y) points on an            ellipse or isosigmachron.        -   (e) plot_tbounds.m: plots the bounds for the primary point            as a function of constellation number for 2-D locations.        -   (f) usedirninfo.m: Uses directional information at receiver            to retain only realizations of (x,y) for the primary point            that are consistent with the directional information.    -   4. boundP_(—)3d.m: Main matlab program. Can do Sequential Bound        Estimation for ellipsoidal or isosigmachronic locations using        constellations of four stations. It uses an analytical solution        for location, and can iterate analytical solution if there are        advective effects. Program has several ascii input files that        supply data and modes for its operation. The program is started        by starting matlab and typing the program name. It calls the        following matlab script files plus some others that have been        listed already:        -   (a) PU_lim.m: Gets the P % confidence limits for the winds            using all constellations.        -   (b) Pcc_lim.m: Gets the P % confidence limits for each speed            of sound between the primary point and each receiver using            all constellations and between the source and primary point.        -   (c) Precpos_lim.m: Gets the P % confidence limits for            coordinates of the receivers using all constellations.        -   (d) Ptxyz.m: Gets the P % confidence limits for each of x,            y, and z for the primary point from each constellation of            receivers.        -   (e) Pxyz.m: Gets the P % confidence limits for each of x, y,            and z for the source from each constellation.        -   (f) chkint_type.m: Checks if the bounds for a parameter            yield at least one physical location from their            intersections.        -   (g) confidlim.m: Computes P percent confidence limits for a            one-dimensional variable, x.        -   (h) funl.m: Used to compute the value of a quartic            polynomial.        -   (i) getrvar.m: Gets random variables for a realization.        -   (j) plot_c.m: plots the bounds for the speed of sound            between the primary point and each receiver and between the            source and the primary point.        -   (k) plt_U.m: This plots the bounds for the spatially            homogeneous winds as a function of constellation number            used.        -   (l) plt_reclim.m: This plots the bounds for the receiver            coordinates as a function of constellation.        -   (m) plt_sourcelim.m: This plots the bounds for the source            location as a function of constellation.        -   (n) plt_tbnds.m: This plots the bounds for the primary point            as a function of constellation.        -   (o) sign_chg.m: Is used to aid in location of finding roots            of polynomials for high precision root finding.        -   (p) sol_quartic_(—isosig)_location.m: Provides solution to            analytical equations for quartic equation. Used for            isosigmachronic location and for ellipsoidal location.    -   5. boundP.m: Can implement Sequential Bound Estimation for        ellipsoidal and isosigmachronic locations in three dimensions        using constellations of three receivers and one source. Program        has several ascii input files that supply data and modes for its        operation. It is a matlab program which can be started by        starting matlab and typing boundP.m. It has several ascii input        files that provide it data and tell it what to do. It calls the        following programs that have not yet been listed above:        -   (a) comp1.m: Finds the compliment of the values in a vector.            Values in the vector can be 0 or 1.        -   (b) ranint.m: Generates N independent uniformly distributed            integers.        -   (c) sol_quartic_isosig_loc.m: Solves quartic or quadratic            equation for isosigmachronic or ellipsoidal location in            three dimensions.    -   6. boundsposit_(—)4rec_d.m: Uses isodiachronic or hyperbolic        location to implement Sequential Bound Estimation. This program        can use a synthetic database of travel time differences to aid        in finding location of the primary point. To run this program,        ascii input files are made containing data and instructions for        the program. Once these files are made, the program is started        by starting matlab, and typing boundsposit_(—)4rec_d.m. It calls        the following new matlab script files that have not been listed        above:        -   (a) removepair.m: Is used to remove receivers that are not            desired for locating the primary point.        -   (b) getsrcbnds.m: Finds spatial bounds for the primary point            from a synthetic database of differences in travel time.        -   (c) sbnds_ttdgroup.m: Called by getsrcbnds.m.        -   (d) unpack_binmatch.m: Called by sbnds_ttdgroup.m.        -   (e) bin_numb_to_coord.m: Called by sbnds_ttdgroup.m.        -   (f) getrv.m: Forms many of the random variables for the            Monte-Carlo realizations.        -   (g) solve_quartic_location.m: Provides solution to            analytical equation for the quartic equation for            isodiachronic geometry. Also provides solution when the            equation involves hyperbolic geometry.        -   (h) plt_sbnds.m: Plots bounds for the location of the            primary point as a function of constellation number.        -   (i) plt_c.m: This plots the bounds for the speed of sound            between the primary point and each station.        -   (j) Pc_lim.m: Gets the P % confidence limits for each speed            of sound between the primary point and each station.

CONCLUSIONS, RAMIFICATIONS, AND SCOPE

Thus the reader will see the utility of using the methods for computingprobability distributions and bounds for many types of locationproblems. The methods are robust and incorporate realistic priordistributions of error including locations of the stations, errors inthe measurements, errors in the speed of propagation, and errors relatedto the models of winds and currents that may significantly affect thedata. The methods do not make the ubiquitous linear approximation forobtaining bounds or probability distributions. Instead, the methodsincorporate nonlinearity of a problem without approximation.

When the speeds of propagation differ between a primary point and atleast two stations, locations from differences in times can be estimatedusing a new geometrical surface called an “isodiachron.” An isodiachronis the locus of points on which the difference in travel time isconstant. It is possible to use the idea of isodiachrons to obtainanalytical solutions for location thus reducing the computational effortrequired to obtain bounds and probability distributions for location.

When the speeds of propagation differ between a primary point and atleast two stations, locations can be made from measured travel timesthat traverse these three points. Analytical solutions for location canmake use of a new geometrical surface called an “isosigmachron.” It isdefined to be the locus of points on which the sum of travel times isconstant. The analytical solutions for location reduce the computationaleffort required to obtain bounds and probability distributions forlocation.

A new method called “Sequential Bound Estimation” is introduced forobtaining bounds or probability distributions of location forhyperbolic, isodiachronic, ellipsoidal, isosigmachronic, or sphericallocation. It has the advantage of being able to handle fully nonlinearrelationships between location and the measurements and other variableswithout making any linearizing approximation. Furthermore, it can handledata or model evolution in a sequential manner without restricting anyparameters or variables to be Gaussian. In fact, the prior distributionsof the measurements and variables can be realistic, which leads to morerealistic estimates than obtained from prior art.

Sequential Bound Estimation can be applied to problems involving dataand variables through a primary function. Such problems arise in manyfields in navigation and location.

Sequential Bound Estimation can be applied in other fields such as thosedealing with the control of systems and machinery, robotics, and bothlinear and nonlinear geophysical inversion methods.

It is important to mention that the methods in this invention apply whenthe locations of a source and receivers are interchanged. When there areno effects from advection, the reciprocity theorem guarantees the fieldat the source is identical to that at a receiver. When advection plays asignificant role, the techniques for locating a primary point are thesame. It is also important to point out that if one estimates traveltimes, differences, or sums of travel time, that the travel times can beobtained in many different ways such as with transponders. The methodsgiven in this invention apply to obtaining the data in different ways.

In the foregoing specification, the estimation methods and locationtechniques have been described with reference to specific exemplaryembodiments thereof. These techniques are considered to have beendescribed in such full, clear, concise and exact terms as to enable aperson of ordinary skill in the art to make and use the same. It will beapparent to those skilled in the art, that a person understanding themethod for obtaining bounds or probability distributions for locationmay conceive of changes or other embodiments or variations, whichutilize the principles of this invention without departing from thebroader spirit and scope of the invention as set forth in the appendedclaims. All are considered within the sphere, spirit, and scope of theinvention. The specification and drawings are, therefore, to be regardedin an illustrative rather than restrictive sense. Accordingly, it is notintended that the invention be limited except as may be necessary inview of the appended claims, which particularly point out and distinctlyclaim the subject matter Applicants regard as their invention.

1. A method to estimate at least one function of a probabilitydistribution of at least one variable, where the variables consist of aprimary function and arguments of said primary function, and where atleast one said argument is a datum, and the data are separated into atleast two groups and processed sequentially, comprising the steps of:(a) assigning prior bounds for said datum, said arguments, and saidprimary function; (b) assigning prior probability distributions withinsaid prior bounds for all said arguments of said primary function; (c)generating realizations of said primary function from realizations ofsaid arguments using Monte-Carlo technique; (d) determining at least oneposterior bound among said realizations of said primary function andsaid arguments; (e) stopping the sequential processing when there areless than a minimum number of acceptable said realizations of saidprimary function within its said prior bounds; (f) stopping saidsequential processing when at least one said posterior bound isinconsistent with at last one said prior bound; (g) determining at leastone final bound after said data are sequentially processed; whereby atleast one said function of said probability distribution of at least onesaid variable is estimated, where said variables consist of said valuesof said primary function and said arguments of said primary function,and where at least one said argument is said datum, and said data areseparated into at least two said groups and processed sequentially.2-57. (canceled)